Chapter 1: Equations and inequalities

Chapter 1:Equations and inequalitiesBIG IDEAS:Use properties to evaluate and simplify expressionsUse problem solving strategies and verbal modelsSolve linear and absolute value equations and inequalitiesWhat is the difference between a daily low temperature of -5 and a daily high temperature of 18?

Lesson 1: Apply Properties of Real

Essential questionHow are addition and subtraction related and how are multiplication and division related?Opposite: additive inverse: the opposite of b is b

Reciprocal: multiplicative inverse: the reciprocal of a = 1/a. VOCABULARY

Properties

EXAMPLE 1Graph real numbers on a number lineGraph the real numbers and 3 on a number line.54SOLUTIONNote that = 1.25. Use a calculator to approximate 3 to the nearest tenth:54

3 1.7. (The symbol means is approximately equal to.)

So, graph between 2 and 1, and graph 3 between1 and 2, as shown on the number line below.54

EXAMPLE 3Identify properties of real numbersIdentify the property that the statement illustrates. 7 + 4 = 4 + 7 13 = 1113

SOLUTION Inverse property of multiplicationCommutative property of additionSOLUTION

EXAMPLE 4Use properties and definitions of operationsUse properties and definitions of operations to show that a + (2 a) = 2. Justify each step.SOLUTIONa + (2 a)= a + [2 + ( a)]Definition of subtraction= a + [( a) + 2]Commutative property of addition= [a + ( a)] + 2Associative property of addition= 0 + 2Inverse property of addition= 2Identity property of addition

Identify the property that the statement illustrates. 15 + 0 = 15SOLUTIONIdentity property of addition.Associative property of multiplication.SOLUTION (2 3) 9 = 2 (3 9)

GUIDED PRACTICEfor Examples 3 and 4

Identify the property that the statement illustrates. 4(5 + 25) = 4(5) + 4(25)SOLUTIONIdentity property of multiplication.Distributive property.SOLUTION 1 500 = 500

GUIDED PRACTICEfor Examples 3 and 4

Use properties and definitions of operations to show that the statement is true. Justify each step.SOLUTIONDef. of division

GUIDED PRACTICEfor Examples 3 and 41b= b ( 4)

Comm. prop. of multiplicationAssoc. prop. of multiplication1b= (b ) 4

= 1 4

Inverse prop. of multiplicationIdentity prop. of multiplication= 41b= b (4 )b (4 b)

b (4 b) = 4 when b = 0

Use properties and definitions of operations to show that the statement is true. Justify each step.SOLUTION 3x + (6 + 4x) = 7x + 6

GUIDED PRACTICEfor Examples 3 and 4Assoc. prop. of additionCombine like terms.Comm. prop. of addition3x + (6 + 4x)= 3x + (4x + 6)= (3x + 4x) + 6= 7x + 6Essential questionHow are addition and subtraction related and how are multiplication and division related?They are inverses of one another. Subtraction is defined as adding the opposite of the number being subtracted. Division is defined as multiplying by the reciprocal of the divisor.Jill has enough money for a total of 32 table decorations and wall decorations. If n is the number of table decorations, write an expression for the number of wall decorations she can buy.

Lesson 2: Evaluate and simplify algebraic expressionEssential questionWhen an expression involves more than one operation, in what order do you do the operations?Power: an expression formed by repeated multiplication of the same factor

Variable: a letter that is used to represent one or more numbers

Term: each part of an expression separated by + and signs

Coefficient: the number that leads a variable

Identity: a statement that equates to two equivalent expressionsVOCABULARY

EXAMPLE 1Evaluate powers (5)4 54

= (5) (5) (5) (5)= 625

= (5 5 5 5)= 62521

EXAMPLE 2Evaluate an algebraic expressionEvaluate 4x2 6x + 11 when x = 3.4x2 6x + 11= 4(3)2 6(3) + 11Substitute 3 for x.= 4(9) 6(3) + 11Evaluate power.= 36 + 18 + 11Multiply.= 7Add.22

EXAMPLE 3Use a verbal model to solve a problemCraft FairYou are selling homemade candles at a craft fair for $3 each. You spend $120 to rent the booth and buy materials for the candles. Write an expression that shows your profit fromselling c candles. Find your profit if you sell 75 candles.23

EXAMPLE 3Use a verbal model to solve a problemSOLUTIONSTEP 1Write: a verbal model. Then write an algebraic expression. Use the fact that profit is the difference between income and expenses.

An expression that shows your profit is 3c 120.3

c12024

EXAMPLE 3Use a verbal model to solve a problemSTEP 2Evaluate: the expression in Step 1 when c = 75.3c 120= 3(75) 120Substitute 75 for c.= 225 120= 105Subtract.ANSWER Your profit is $105.Multiply.25

GUIDED PRACTICEfor Examples 1, 2, and 3Evaluate the expression.63 26 SOLUTION216SOLUTION 6426

GUIDED PRACTICEfor Examples 1, 2, and 3(2)6 5x(x 2) when x = 6 SOLUTION 64SOLUTION12027

GUIDED PRACTICEfor Examples 1, 2, and 33y2 4y when y = 2 SOLUTION20(z + 3)3 when z = 1 SOLUTION 6428

GUIDED PRACTICEfor Examples 1, 2, and 3What If? In Example 3, find your profit if you sell 135 candles. ANSWER Your profit is $285.29

EXAMPLE 4Simplify by combining like terms 8x + 3x= (8 + 3)xDistributive property= 11xAdd coefficients. 5p2 + p 2p2= (5p2 2p2) + pGroup like terms.= 3p2 + pCombine like terms. 3(y + 2) 4(y 7)= 3y + 6 4y + 28Distributive property= (3y 4y) + (6 + 28)Group like terms.= y + 34Combine like terms.30

EXAMPLE 4Simplify by combining like terms 2x 3y 9x + y= (2x 9x) + ( 3y + y)Group like terms.= 7x 2yCombine like terms.31

GUIDED PRACTICEfor Example 58. Identify the terms, coefficients, like terms, and constant terms in the expression 2 + 5x 6x2 + 7x 3. Then simplify the expression.SOLUTIONTerms: Coefficients:Like terms:Constants:6x2 +12x 1Simplify:2, 5x, 6x2 , 7x, 35 from 5x, 6 from 6x2 , 7 from 7x5x and 7x, 2 and 32 and 332

GUIDED PRACTICEfor Example 5 15m 9mSOLUTION 6mSimplify the expression. 2n 1 + 6n + 5SOLUTION8n + 433

GUIDED PRACTICEfor Example 5 3p3 + 5p2 p3SOLUTION2p3 + 5p2 2q2 + q 7q 5q2SOLUTION3q2 6q34

GUIDED PRACTICEfor Example 5 8(x 3) 2(x + 6)SOLUTION6x 36 4y x + 10x + ySOLUTION9x 3y 35Essential questionWhen an expression involves more than one operation, in what order do you do the operations?Order of operations: Parenthesis, Exponents, Multiplication, Division, Addition, SubtractionOn a blank sheet of paper, complete #1-13 ODD on P16 in the blue Quiz section. Please turn into the homework bin when finished.

Lesson 3: solve linear equationsEssential questionWhat are the steps for solving a linear equation?Equation: a statement that two expressions are equal

Linear equation: may be written in the form ax + b = 0; no exponents

Solution: a number that makes a true statement when substituted into an equation

Equivalent equations: two equations that have the same solutionsVOCABULARY

EXAMPLE 1Solve an equation with a variable on one sideSolve 45x + 8 = 20. 45x + 8 = 2045x = 12x = (12)54x = 15Write original equation.Subtract 8 from each side.Multiply each side by , the reciprocal of .5445Simplify.ANSWERThe solution is 15.CHECK x = 15 in the original equation.4545x + 8 = (15) + 8 = 12 + 8 = 2041

EXAMPLE 2Write and use a linear equationDuring one shift, a waiter earns wages of $30 and gets an additional 15% in tips on customers food bills. The waiter earns $105. What is the total of the customers food bills?RestaurantSOLUTIONWrite a verbal model. Then write an equation. Write 15% as a decimal.

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EXAMPLE 2Write and use a linear equation105 = 30 + 0.15x75 = 0.15x500 = xWrite equation.Subtract 30 from each side.Divide each side by 0.15.The total of the customers food bills is $500.ANSWER43

GUIDED PRACTICEfor Examples 1 and 2Solve the equation. Check your solution.1. 4x + 9 = 21ANSWERThe solution is x = 3.2. 7x 41 = 13ANSWERThe solution is x = 4.ANSWERThe solution is 5.3.35x + 1 = 444

GUIDED PRACTICEfor Examples 1 and 2A real estate agents base salary is $22,000 per year. The agent earns a 4% commission on total sales. How much must the agent sell to earn $60,000 in one year?4.REAL ESTATEThe agent must sell $950,000 in a year to each $ 60000ANSWER45

EXAMPLE 3Standardized Test Practice

SOLUTION7p + 13 = 9p 513 = 2p 518 = 2p9 = pWrite original equation.Subtract 7p from each side.Add 5 to each side.Divide each side by 2.ANSWERThe correct answer is D

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EXAMPLE 3Standardized Test PracticeCHECK7p + 13 = 9p 57(9) + 13 9(9) 5=?63 + 13 81 5=?76 = 76 Write original equation.Substitute 9 for p.Multiply.Solution checks.47

EXAMPLE 4Solve an equation using the distributive propertySolve 3(5x 8) = 2(x + 7) 12x.3(5x 8) = 2(x + 7) 12x15x 24 = 2x 14 12x15x 24 = 10x 1425x 24 = 1425x = 10x =25Write original equation.Distributive propertyCombine like terms.Add 10x to each side.Add 24 to each side.Divide each side by 25 and simplify.ANSWERThe solution2548

EXAMPLE 4Solve an equation using the distributive propertyCHECK25 3(5 8) 2( + 7) 12 2525

=?

3(6) 14 45=?245 18 = 1825Substitute for x.Simplify.Solution checks.49

EXAMPLE 5Solve a work problemCar WashIt takes you 8 minutes to wash a car and it takes a friend 6 minutes to wash a car. How long does it take the two of you to wash 7 cars if you work together?SOLUTIONSTEP 1Write a verbal model. Then write an equation.

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EXAMPLE 5Solve a work problem

Solve the equation for t.STEP 218t + t = 71624( t + t) = 24 (7)18163t + 4t = 1687t = 168t = 24Write equation.Multiply each side by the LCD, 24.Distributive propertyCombine like terms.Divide each side by 7.ANSWERIt will take 24 minutes to wash 7 cars if you work together.51

EXAMPLE 5Solve a work problemCHECKYou wash 24 = 3 cars and your friend washes 24 = 4 cars in 24 minutes. Together, you wash 7 cars.1618

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GUIDED PRACTICEfor Examples 3, 4, and 5Solve the equation. Check your solution.5. 2x + 9 = 2x 7ANSWERThe correct answer is 4.6. 10 x = 6x + 15ANSWERThe correct answer is 1.7. 3(x + 2) = 5(x + 4)ANSWERThe solution is 7.53

GUIDED PRACTICEfor Examples 3, 4, and 5Solve the equation. Check your solution.8. 4(2x + 5) = 2(x 9) 4xANSWERThe solution x = 1x + x = 3914259.ANSWERThe correct answer is 6054

GUIDED PRACTICEfor Examples 3, 4, and 5Solve the equation. Check your solution.10. x + = x 122356ANSWERThe correct answer is 4What If? In Example 5, suppose it takes you 9 minutes to wash a car and it takes your friend 12 minutes to wash a car. How long does it take the two of you to wash 7 cars if you work together?11.ANSWERIt will take 36 minutes to wash 7 cars if you work together.55Essential questionWhat are the steps for solving a linear equation?If the equation involves an expression in parenthesis, remove the parentheses by using the distributive property. Then use the properties of equality to obtain equivalent equations in a series of steps until you obtain an equation of the form x = a.Use the distributive property to rewrite xy-5y as a product.

Lesson 4: Rewrite formulas and equationsEssential questionWhat are formulas, and how are formulas used?Formula: an equation that relates two or more quantities, usually represented by variables

Solve for a variable: to rewrite an equation as an equivalent equation in which the variable is on one side and does not appear on the other side; isolate the variableVOCABULARY

EXAMPLE 1Rewrite a formula with two variablesSolve the formula C = 2r for r. Then find the radius of a circle with a circumference of 44 inches.SOLUTIONC = 2rC2= rSTEP 1Solve the formula for r.STEP 2Substitute the given value into the rewritten formula.Write circumference formula.Divide each side by 2.r =C2=442

7Substitute 44 for C and simplify.The radius of the circle is about 7 inches.ANSWER61

GUIDED PRACTICEfor Example 1Find the radius of a circle with a circumference of 25 feet.1.The radius of the circle is about 4 feet.ANSWER62

GUIDED PRACTICEfor Example 1The formula for the distance d between opposite vertices of a regular hexagon is d = where a is the distance between opposite sides. Solve the formula for a. Then find a when d = 10 centimeters.2.

2a3SOLUTIONd 3a =235When d = 10cm, a = or 8.7cm

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EXAMPLE 2Rewrite a formula with three variablesSOLUTIONSolve the formula for w.STEP 1P = 2l + 2wP 2l = 2wP 2l2= wWrite perimeter formula.Subtract 2l from each side.Divide each side by 2.Solve the formula P = 2l + w for w. Then find the width of a rectangle with a length of 12 meters and a perimeter of 41 meters.

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EXAMPLE 2Rewrite a formula with three variables41 2(12)2w =w = 8.5Substitute 41 for P and 12 for l.Simplify.The width of the rectangle is 8.5 meters.ANSWERSubstitute the given values into the rewritten formula.STEP 265

GUIDED PRACTICEfor Example 2Solve the formula P = 2l + 2w for l. Then find the length of a rectangle with a width of 7 inches and a perimeter of 30 inches.3.Length of rectangle is 8 in.ANSWERSolve the formula A = lw for w. Then find the width of a rectangle with a length of 16 meters and an area of 40 square meters.4.Write of rectangle is 2.5 mw = AlANSWER66

GUIDED PRACTICEfor Example 2Solve the formula for the variable in red. Then use the given information to find the value of the variable.A =12bh5.

Find h if b = 12 mand A = 84 m2. =h2A bANSWER67

GUIDED PRACTICEfor Example 2Find the value of h if b = 12m and A = 84m2.

Find h if b = 12 mand A = 84 m2. =h2A bh = 14mANSWER68

GUIDED PRACTICEfor Example 2Find b if h = 3 cmSolve the formula for the variable in red. Then use the given information to find the value of the variable.A =12bh6.

and A = 9 cm2. =b2A hANSWER69

GUIDED PRACTICEfor Example 2Find b if h = 3 cmSolve the formula for the variable in red. Then use the given information to find the value of the variable.A =12bh6.

and A = 9 cm2.b = 6cmANSWER70

GUIDED PRACTICEfor Example 2Solve the formula for the variable in red. Then use the given information to find the value of the variable.A =127.(b1 + b2)h

Find h if b1 = 6 in.,b2 = 8 in., and A = 70 in.2h =2A(b1 + b2)ANSWER71

GUIDED PRACTICEfor Example 2Solve the formula for the variable in red. Then use the given information to find the value of the variable.A =127.(b1 + b2)h

Find h if b1 = 6 in.,b2 = 8 in., and A = 70 in.2 h = 10 in.ANSWER72

EXAMPLE 3Rewrite a linear equation Solve 9x 4y = 7 for y. Then find the value of y when x = 5.SOLUTIONSolve the equation for y.STEP 1 9x 4y = 74y = 7 9xy =9474+xWrite original equation.Subtract 9x from each side.Divide each side by 4.73

EXAMPLE 3Rewrite a linear equation Substitute the given value into the rewritten equation.STEP 2y =9474+ (5)y =45474y = 13CHECK9x 4y = 7 9(5) 4(13) 7=?7 = 7 Substitute 5 for x.Multiply.Simplify.Write original equation.Substitute 5 for x and 13 for y.Solution checks.74

EXAMPLE 4Rewrite a nonlinear equation Solve 2y + xy = 6 for y. Then find the value of y when x = 3.SOLUTIONSolve the equation for y.STEP 12y + x y = 6(2+ x) y = 6y =62 + xWrite original equation.Distributive propertyDivide each side by (2 + x).75

EXAMPLE 4Rewrite a nonlinear equation Substitute the given value into the rewritten equation.STEP 2y =62 + (3)y = 6Substitute 3 for x.Simplify.76

GUIDED PRACTICEfor Examples 3 and 4Solve the equation for y. Then find the value of y when x = 2.8. y 6x = 7y = 7 + 6xy = 19ANSWER 9. 5y x = 13y =x5135+y = 5ANSWER 10. 3x + 2y = 12y = 3x2+ 6ANSWER y = 377

GUIDED PRACTICEfor Examples 3 and 4Solve the equation for y. Then find the value of y when x = 2.11. 2x + 5y = 112. 3 = 2xy x13. 4y xy = 28y = 14 284 xy =ANSWER y = 1413 +x 2xy =ANSWER 2x515y = 1y =ANSWER 78Essential questionWhat are formulas, and how are formulas used?Formulas are equations that relate two or more quantities, usually represented by variables. Formulas can be used to solve many real-world problems, such as problems about investment, temperature, perimeter, area and volume.A balloon is released from a height of 5 feet above the ground. Its altitude (in feet) after t minutes is given by the expression 5+82t. What is the altitude of the balloon after 6 minutes.

Lesson 5: use problem solving strategies and modelsEssential questionHow can problem solving strategies be used to find verbal and algebraic models?Verbal model: a word equation that may be written before an equation is written in mathematical symbolsVOCABULARY

EXAMPLE 1Use a formulaHigh-speed TrainThe Acela train travels between Boston and Washington, a distance of 457 miles. The trip takes 6.5 hours. What is the average speed?SOLUTIONYou can use the formula for distance traveled as a verbal model.457= r6.5

Distance (miles)= Rate (miles/hour) Time (hours)

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EXAMPLE 1Use a formulaAn equation for this situation is 457 = 6.5r. Solve for r.457 = 6.5r

70.3rWrite equation.Divide each side by 6.5.The average speed of the train is about 70.3 miles per hour.ANSWERYou can use unit analysis to check your answer.457 miles6.5 hours70.3 miles1 hour

CHECK85

GUIDED PRACTICEfor Example 11. AVIATION: A jet flies at an average speed of 540 miles per hour. How long will it take to fly from New York to Tokyo, a distance of 6760 miles?Jet takes about 12.5 hours to fly from New York to Tokyo.ANSWER86

EXAMPLE 2Look for a patternParamotoringA paramotor is a parachute propelled by a fan-like motor. The table shows the height h of a paramotorist t minutes after beginning a descent. Find the height of the paramotorist after 7 minutes.

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EXAMPLE 2Look for a patternSOLUTION

The height decreases by 250 feet per minute.You can use this pattern to write a verbal model for the height.An equation for the height is h = 2000 250t.88

EXAMPLE 2Look for a patternSo, the height after 7 minutes is h = 2000 250(7) = 250 feet.ANSWER89

EXAMPLE 3Draw a diagramBannersYou are hanging four championship banners on a wall in your schools gym. The banners are 8 feet wide. The wall is 62 feet long. There should be an equal amount of space between the ends of the wall and the banners, and between each pair of banners. How far apart should the banners be placed?SOLUTIONBegin by drawing and labeling a diagram, as shown below.

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EXAMPLE 3Draw a diagramFrom the diagram, you can write and solve an equation to find x.x + 8 + x + 8 + x + 8 + x + 8 + x=625x + 32=62Subtract 32 from each side.5x=30x=6Divide each side by 5.Combine like terms.Write equation.The banners should be placed 6 feet apart.ANSWER91

EXAMPLE 4Standardized Test Practice

SOLUTIONSTEP 1 Write a verbal model. Then write an equation.

An equation for the situation is 460 = 30g + 25(16 g).92

EXAMPLE 4Standardized Test PracticeSolve for g to find the number of gallons used on the highway.STEP 2 460 = 30g + 25(16 g)460 = 30g + 400 25g460 = 5g + 400 60 = 5g 12 = gWrite equation.Distributive propertyCombine like terms.Subtract 400 from each side.Divide each side by 5.The car used 12 gallons on the highway.ANSWERThe correct answer is B.

CHECK:30 12 + 25(16 12)

= 360 + 100

= 46093

GUIDED PRACTICEfor Examples 2, 3 and 42. PARAMOTORING: The table shows the height h of a paramotorist after t minutes. Find the height of the paramotorist after 8 minutes.

So, the height after 8 minutes is h = 2400 210(8) = 720 ftANSWER94

GUIDED PRACTICEfor Examples 2, 3 and 43. WHAT IF? In Example 3, how would your answer change if there were only three championship banners?The space between the banner and walls and between each pair of banners would increase to 9.5 feet.ANSWER95

GUIDED PRACTICEfor Examples 2, 3 and 44. FUEL EFFICIENCY A truck used 28 gallons of gasoline and traveled a total distance of 428 miles. The trucks fuel efficiency is 16 miles per gallon on the highway and 12 miles per gallon in the city. How many gallons of gasoline were used in the city?Five gallons of gas were used.ANSWER96Essential questionHow can problem solving strategies be used to find verbal and algebraic models?The problem solving strategies use a formula and look for a pattern that can be used to write verbal models which can then be used to write algebraic models. The strategy draw a diagram can be used to write an algebraic model directly.On a blank sheet of paper complete #2-12 Even on P40 in the blue quiz section. Turn into the homework bin when finished.

Lesson 6: solve linear inequalitiesEssential questionHow are the rules for solving linear inequalities similar to those for solving linear equations, and how are they different?Linear inequality: an inequality using , ,

Compound inequality: consists of two simple inequalities joined by and or or

Equivalent inequalities: inequalities that have the same solutions as the original inequalityVOCABULARY Solve inequalities just the same as equalities using the Order of Operations

When multiplying or dividing by a negative, flip the inequality sign. Note:

EXAMPLE 1Graph simple inequalitiesa. Graph x < 2.The solutions are all real numbersless than 2.

An open dot is used in the graph to indicate 2 is not a solution.103

EXAMPLE 1Graph simple inequalitiesb. Graph x 1.The solutions are all real numbers greater than or equal to 1.A solid dot is used in the graph to indicate 1 is a solution.

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EXAMPLE 2Graph compound inequalitiesa. Graph 1 < x < 2.The solutions are all real numbers that are greater than 1 and less than 2.

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EXAMPLE 2Graph compound inequalitiesb. Graph x 2 or x > 1.The solutions are all real numbers that are less than or equal to 2 or greater than 1.

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GUIDED PRACTICEfor Examples 1 and 2Graph the inequality.1. x > 5The solutions are all real numbers greater than 5.An open dot is used in the graph to indicate 5 is not a solution.

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GUIDED PRACTICEfor Examples 1 and 2Graph the inequality.2. x 3The solutions are all real numbers less than or equal to 3.A closed dot is used in the graph to indicate 3 is a solution.

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GUIDED PRACTICEfor Examples 1 and 2Graph the inequality.3. 3 x < 1The solutions are all real numbers that are greater than or equalt to 3 and less than 1.

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GUIDED PRACTICEfor Examples 1 and 2Graph the inequality.4. x < 1 or x 2The solutions are all real numbers that are less than 1 or greater than or equal to 2.

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EXAMPLE 3Solve an inequality with a variable on one sideFairYou have $50 to spend at a county fair. You spend $20 for admission. You want to play a game that costs $1.50. Describe the possible numbers of times you can play the game.SOLUTIONSTEP 1Write a verbal model. Then write an inequality.

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EXAMPLE 3Solve an inequality with a variable on one sideAn inequality is 20 + 1.5g 50.STEP 2Solve the inequality.20 + 1.5g 501.5g 30g 20Write inequality.Subtract 20 from each side.Divide each side by 1.5.ANSWERYou can play the game 20 times or fewer.112

EXAMPLE 4Solve an inequality with a variable on both sidesSolve 5x + 2 > 7x 4. Then graph the solution.5x + 2 > 7x 4 2x + 2 > 4 2x > 6x < 3Write original inequality.Subtract 7x from each side. Subtract 2 from each side.Divide each side by 2 and reverse the inequality.ANSWERThe solutions are all real numbers less than 3. The graph is shown below.

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GUIDED PRACTICEfor Examples 3 and 4Solve the inequality. Then graph the solution.5. 4x + 9 < 256. 1 3x 147. 5x 7 6x8. 3 x > x 9x < 4ANSWER x 5ANSWER x < 6ANSWER x > 7 ANSWER 114

EXAMPLE 5Solve an and compound inequalitySolve 4 < 6x 10 14. Then graph the solution. 4 < 6x 10 14 4 + 10 < 6x 10 + 10 14 + 106 < 6x 241 < x 4Write original inequality.Add 10 to each expression.Simplify.Divide each expression by 6.ANSWERThe solutions are all real numbers greater than 1 and less than or equal to 4. The graph is shown below.

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EXAMPLE 6Solve an or compound inequalitySolve 3x + 5 11 or 5x 7 23. Then graph the solution.SOLUTIONA solution of this compound inequality is a solution of either of its parts.First InequalitySecond Inequality3x + 5 113x 6x 2Write first inequality.Subtract 5 from each side.Divide each side by 3.5x 7 235x 30x 6Write second inequality.Add 7 to each side.Divide each side by 5.116

EXAMPLE 6Solve an or compound inequalityANSWERThe graph is shown below. The solutions are all real numbers less than or equal to 2 or greater than or equal to 6.

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EXAMPLE 7Write and use a compound inequalityBiology

A monitor lizard has a temperature that ranges from 18C to 34C. Write the range of temperatures as a compound inequality. Then write an inequality giving the temperature range in degrees Fahrenheit.118

EXAMPLE 7Write and use a compound inequalitySOLUTIONThe range of temperatures C can be represented by the inequality 18 C 34. Let F represent the temperature in degrees Fahrenheit.18 C 34Write inequality.18 3459(F 32)32.4 F 32 61.264.4 F 93.2 Substitute for C.95 (F 32) Multiply each expression by , the reciprocal of .9559Add 32 to each expression.119

EXAMPLE 7Write and use a compound inequalityANSWERThe temperature of the monitor lizard ranges from 64.4F to 93.2F.120

GUIDED PRACTICEfor Examples 5,6, and 7Solve the inequality. Then graph the solution.9. 1 < 2x + 7 < 19ANSWERThe solutions are all real numbers greater than 4 and less than 6.

4 < x < 6121

GUIDED PRACTICEfor Examples 5,6 and 7Solve the inequality. Then graph the solution.10. 8 x 5 6The solutions are all real numbers greater than and equal to 11 and less than and equal to 3.ANSWER

11 x 3122

GUIDED PRACTICEfor Examples 5,6 and 7Solve the inequality. Then graph the solution.11. x + 4 9 or x 3 7ANSWERThe graph is shown below. The solutions are all real numbers.less than or equal to 5 or greater than or equal to 10.

x 5 or x 10123

GUIDED PRACTICEfor Examples 5,6 and 7Solve the inequality. Then graph the solution.12. 3x 1< 1 or 2x + 5 11 x < 0 or x 3less than 0 or greater than or equal to 3.ANSWERThe graph is shown below. The solutions are all real numbers.

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GUIDED PRACTICEfor Examples 5,6 and 713.WHAT IF? In Example 7, write a compound inequality for a lizard whose temperature ranges from 15C to 30C. Then write an inequality giving the temperature range in degrees Fahrenheit.15 C 30 or 59 F 86ANSWER 125Essential questionHow are the rules for solving linear inequalities similar to those for solving linear equations, and how are they different?The addition and subtraction properties are the same, but if you multiply or divide both sides of an inequality by a negative number, the inequality symbol must be reversed.What is unique about absolute value numbers that is not true of integers?

Lesson 7: Solve absolute value equations and inequalitiesEssential questionHow are absolute value equations and inequalities like linear equations and inequalities?Absolute value: the distance an umber is from 0 on a number line; always positive

Extraneous solution: an apparent solution that must be rejected because it does not satisfy the original equationVOCABULARY

EXAMPLE 1Solve a simple absolute value equationSolve |x 5| = 7. Graph the solution.SOLUTION| x 5 | = 7x 5 = 7 or x 5 = 7 x = 5 7 or x = 5 + 7 x = 2 or x = 12 Write original equation.Write equivalent equations.Solve for x.Simplify.131

EXAMPLE 1

The solutions are 2 and 12. These are the values of x that are 7 units away from 5 on a number line. The graph is shown below.ANSWERSolve a simple absolute value equation132

EXAMPLE 2

Solve an absolute value equation| 5x 10 | = 455x 10 = 45 or 5x 10 = 45 5x = 55 or 5x = 35 x = 11 or x = 7 Write original equation.Expression can equal 45 or 45 .Add 10 to each side.Divide each side by 5.Solve |5x 10 | = 45. SOLUTION133

EXAMPLE 2

Solve an absolute value equationThe solutions are 11 and 7. Check these in the original equation. ANSWERCheck:| 5x 10 | = 45| 5(11) 10 | = 45?|45| = 45?45 = 45

| 5x 10 | = 45 | 5(7) 10 | = 45?45 = 45

| 45| = 45?134

EXAMPLE 3

| 2x + 12 | = 4x 2x + 12 = 4x or 2x + 12 = 4x 12 = 2x or 12 = 6x 6 = x or 2 = x Write original equation.Expression can equal 4x or 4 x Add 2x to each side.Solve |2x + 12 | = 4x. Check for extraneous solutions.SOLUTIONSolve for x.Check for extraneous solutions135

EXAMPLE 3

| 2x + 12 | = 4x | 2(2) +12 | = 4(2)?|8| = 8? 8 = 8

Check the apparent solutions to see if either is extraneous.Check for extraneous solutions| 2x + 12 | = 4x| 2(6) +12 | = 4(6)?|24| = 24? 24 = 24

The solution is 6. Reject 2 because it is an extraneous solution.ANSWERCHECK136

GUIDED PRACTICESolve the equation. Check for extraneous solutions.1. | x | = 5for Examples 1, 2 and 3The solutions are 5 and 5. These are the values of x that are 5 units away from 0 on a number line. The graph is shown below.ANSWER 3 4 2 1 0 1 2 3 4 5 6 7 5 6 7

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GUIDED PRACTICESolve the equation. Check for extraneous solutions.2. |x 3| = 10for Examples 1, 2 and 3The solutions are 7 and 13. These are the values of x that are 10 units away from 3 on a number line. The graph is shown below.ANSWER 3 4 2 1 0 1 2 3 4 5 6 7 5 6 7 8 9 10 11 12 13

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GUIDED PRACTICESolve the equation. Check for extraneous solutions.3. |x + 2| = 7for Examples 1, 2 and 3The solutions are 9 and 5. These are the values of x that are 7 units away from 2 on a number line. ANSWER139

GUIDED PRACTICESolve the equation. Check for extraneous solutions.4. |3x 2| = 13for Examples 1, 2 and 3ANSWERThe solutions are 5 and .140

GUIDED PRACTICESolve the equation. Check for extraneous solutions.5. |2x + 5| = 3xfor Examples 1, 2 and 3The solution of is 5. Reject 1 because it is an extraneous solution.ANSWER141

GUIDED PRACTICESolve the equation. Check for extraneous solutions.6. |4x 1| = 2x + 9for Examples 1, 2 and 3ANSWERThe solutions are and 5. 311142

EXAMPLE 4Solve an inequality of the form |ax + b| > cSolve |4x + 5| > 13. Then graph the solution.SOLUTIONFirst InequalitySecond Inequality4x + 5 < 13 4x + 5 > 13 4x < 184x > 8 x < 92 x > 2Write inequalities.Subtract 5 from each side.Divide each side by 4.The absolute value inequality is equivalent to4x +5 < 13 or 4x + 5 > 13.143

EXAMPLE 4

ANSWERSolve an inequality of the form |ax + b| > c The solutions are all real numbers less than or greater than 2. The graph is shown below. 9 2

144

GUIDED PRACTICEfor Example 4Solve the inequality. Then graph the solution.7. |x + 4| 6

x < 10 or x > 2The graph is shown below.ANSWER145

GUIDED PRACTICEfor Example 4Solve the inequality. Then graph the solution.8. |2x 7|>1ANSWER

x < 3 or x > 4The graph is shown below.146

GUIDED PRACTICEfor Example 4Solve the inequality. Then graph the solution.9. |3x + 5| 10ANSWER

x < 5 or x > 123The graph is shown below.147

EXAMPLE 5Solve an inequality of the form |ax + b| c

A professional baseball should weigh 5.125 ounces, with a tolerance of 0.125 ounce. Write and solve an absolute value inequality that describes the acceptable weights for a baseball.BaseballSOLUTIONWrite a verbal model. Then write an inequality.STEP 1148

EXAMPLE 5

Solve an inequality of the form |ax + b| cSTEP 2Solve the inequality.Write inequality.Write equivalent compound inequality.Add 5.125 to each expression.|w 5.125| 0.125 0.125 w 5.125 0.1255 w 5.25

So, a baseball should weigh between 5 ounces and 5.25 ounces, inclusive. The graph is shown below.ANSWER149

EXAMPLE 6

The thickness of the mats used in the rings, parallel bars, and vault events must be between 7.5 inches and 8.25 inches, inclusive. Write an absolute value inequality describing the acceptable mat thicknesses.Gymnastics

SOLUTIONSTEP 1Calculate the mean of the extreme mat thicknesses.Write a range as an absolute value inequality150

EXAMPLE 6

Mean of extremes = = 7.875 7.5 + 8.25 2Find the tolerance by subtracting the mean from the upper extreme.STEP 2Tolerance = 8.25 7.875Write a range as an absolute value inequality= 0.375151

EXAMPLE 6

STEP 3Write a verbal model. Then write an inequality.A mat is acceptable if its thickness t satisfies |t 7.875| 0.375.ANSWERWrite a range as an absolute value inequality152

GUIDED PRACTICEfor Examples 5 and 6Solve the inequality. Then graph the solution.10. |x + 2| < 6The solutions are all real numbers less than 8 or greater than 4. The graph is shown below.ANSWER

8 < x < 4153

GUIDED PRACTICEfor Examples 5 and 6Solve the inequality. Then graph the solution.11. |2x + 1| 9The solutions are all real numbers less than 5 or greater than 4. The graph is shown below.ANSWER

5 x 4154

GUIDED PRACTICEfor Examples 5 and 612. |7 x| 4Solve the inequality. Then graph the solution.3 x 11ANSWERThe solutions are all real numbers less than 3 or greater than 11. The graph is shown below.

155

GUIDED PRACTICEfor Examples 5 and 613. Gymnastics: For Example 6, write an absolute value inequality describing the unacceptable mat thicknesses.A mat is unacceptable if its thickness t satisfies |t 7.875| > 0.375.ANSWER156Essential questionHow are absolute value equations and inequalities like linear equations and inequalities?An absolute value equation can be rewritten as two linear equations, and an absolute value inequality can be rewritten as two linear inequalities.