Chapter 6

Solving and Graphing Inequalities

Rules for graphing your answersIf the letter is on the left then we can follow

the direction of the arrowWe must mark the numbers

< or > we use an open circle< or > we use a closed circle

Our answers will look like this and we will graph the answers on a number line

1. X < 22. X > -23. Z > 14. 0 < d

Now you practice:1. T > 12. X >-13. N < 04. 4 >y

Our problems will look like this:

1. X + 5 > 3

We will use our Chapter 3 rules to solve:1. Draw tracks2. Count variables

1. If one, numbers jump tracks

2. If 2 or more where are they1. Same side, family

members2. Different sides, letters

jump3. Add or subtract the

numbers4. Is there a number on the

letters? 1. If its an integer divide

each side2. If it’s a fraction, flip and

multiply3. if the integer or fraction

is negative I must turn the arrow around to the opposite direction

5. Graph the solution

More examples:1. X + 4 < 72. N + 6 > 23. 5 > a + 54. -2 > n – 45. X – 5 > 26. P – 1 < -47. -3 < y - 2

Pg 327 #’s 42 – 54 even

Sec 6.2Solving Equations using Multi or DivWe must be able to identify if there is an

integer on the letter or a fraction on the letter.

If the integer or the fraction is negative then we MUST change the direction of the arrow in our answer

Examples1. a/4 < 42. 4x > 203. k/4 < ½4. 18 < 2k5. 6 < t/56. -21 < 3y

Examples: Negative numbers1. -1/2 y < 52. -12m > 183. -8x < 204. -1/5 p > 15. -2/3 x < -56. -24 < 6t

Word Problems1. Kayla wants to buy some posters for her

dorm room. Posters are on sale for $6 each. Write and solve an inequality to determine how many posters she can buy and spend no more than $25.

2. Crandell plans to take figure skating lessons. He can rent skates for $5 per lesson. He can buy skates for $75. For what number of lessons is it cheaper for him to buy rather than rent skates?

Pg 334 #’s 36 – 46 even

QuizSections 6.1 and 6.2

1.X – 4 < -52.-11 > y + 43.-1/2 x > -54.-3x < -275.-3/4 x < -1/4

Sec 6.3; Solving Multi-step Inequalities

1. 2y – 5 < 72. 5 – x > 43. 3(x + 2) <

74. -2(x + 1) <

25. 2x – 3 > 4x

– 1

Sec 6.3; Solving Multi-step Inequalities

1. 5n – 21 < 8n2. -3z + 15 > 2z3. X + 3 > 2x – 44. 4y – 3 < -y + 12

Word ProblemsYou plan to make and sell candles. You pay

$12 for instructions. The materials for each candle cost $0.50. You plan to sell each candle for $2. Let x be the numbers of candles you sell. How many candles must be sold to make at least $300 profit.

Sec 6.4; Solving compound inequalitiesHow do you combine two thoughts in

English class?What are they called?What words do we use?

How to Solve:1. -2 < x + 2

< 42. -1 < x + 3

< 73. -6 < -3x <

124. 0 < x – 4 <

12

These are “AND” problems. These problems must be re-written into two problems.

Word Problems1. In the summer it took a Pony Express rider

about 10 days to ride from St. Joseph, Missouri to Sacramento, California. In winter it took as many as 16 days. Write an inequality to describe the number of days that the trip might have taken.

Word ProblemsFrequency is used to describe the pitch of a

sound. Frequencies are measured in hertz. Write an inequality for the followingSound of a human voice: 85 hertz to 1100

hertzSound of a bats signal: 10,000 hertz to

120,000 hertzSound heard by a dog: 15 hertz to 50,000

hertzSound heard by a dolphin: 150 hertz to

150,000 hertz

Pg 346 #’s 30-46 even

Sec 6.5; Solving Compound InequalitiesThese are called “OR” problemsThese problems are already written and

ready to be solved

Examples1. X-4<3 or 2x> 182. 3x + 1 < 4 or 2x – 5 > 73. X + 5 < -6 or 3x > 124. 6x – 5 < 7 or 8x + 1 > 25

Word ProblemA baseball is hit straight up in the air. Its

initial velocity is 64 ft per second. Its formula is v = -32t + 64. Find the values of “t” for which the velocity of the baseball is greater than 32 or less than -32 feet per second.

Sec 6.6; Absolute Value EquationsThese problems will create some re-writing

1st – make sure the absolute value bars are on a side by themselves

2nd – drop the bars and write two problems3rd – in the second problem you will need the

opposite of the symbol (equals or inequality) and the opposite of the number.

Now solve the equationsIf its an inequality then you will also graph

the solutions.

For examples we will use p. 356

Pg. 358 #”s 16-26 evenpg 359 #’s 32-40 even

Sec 6.7; Solving absolute value inequalitiesWe will use the same rules as the previous

section (6.6)Make sure to remember to make the

changes to the second problem that you re-write.

We will use pg 364 and 365 for examples

Sec 6.8; Graphing Linear Inequalities in Two VariablesWe use our previous knowledge from

Chapter’s 4 and 5.

To graph we will use y= mx + bSlope intercept form: y = mx + b

“b” is the y-intercept. I must always use this number first when graphing. It is always located on the y-axis, either above or below the origin.

“m” is the slope. Its always a fraction and remember to use rise over run

To graph given when given an equation; turn it into y=mx+bIf the symbol is < or

> we will draw a dashed line

If the symbol is < or > we will draw a normal line

To shade will have to use a test point and the slope intercept form

If the test result is false shade away from the TP, if the test is true shade to the TP.

Flow ChartLet x jump the

tracksIf there is a

number on y, then we will set up three fractions and divide or reduce

Collect “b” and find it on the y-axis

Collect “m” and use rise over run to get to the next point on my line.

-2x + y < 3

MORE EXAMPLES:1. X < -22. Y < 13. X + Y > 34. 2X – Y > -25. 3X – Y < 4

P 371 #’S 26-30 EVEN

P 371 #’S 36-50 EVEN