2.3 Solving Word Problems. Goals SWBAT solve linear inequalities SWBAT solve linear inequalities...

2.3 Solving Word 2.3 Solving Word ProblemsProblems

GoalsGoals

SWBAT solve linear inequalitiesSWBAT solve linear inequalities

SWBAT solve compound SWBAT solve compound inequalitiesinequalities

Solving Real World ProblemsSolving Real World Problems

1.1. Carefully read the problem and Carefully read the problem and decide what the problem is asking decide what the problem is asking for.for.

2.2. Choose a variable to represent one of Choose a variable to represent one of the unknown values.the unknown values.

3.3. Write an equation(s) to represent the Write an equation(s) to represent the relationship(s) stated in the problem. relationship(s) stated in the problem. You may also need to draw a picture.You may also need to draw a picture.

4.4. Solve the equation.Solve the equation.5.5. Check to see that your solution Check to see that your solution

answers the question, if not, be sure answers the question, if not, be sure to answer all parts.to answer all parts.

1. A landscaper has determined 1. A landscaper has determined that together 1 small bag of that together 1 small bag of lawn seed and 3 large bags will lawn seed and 3 large bags will cover 330 m2 of ground. If the cover 330 m2 of ground. If the large bag covers 50 m2 more large bag covers 50 m2 more than the small bag, what is the than the small bag, what is the area covered by each size bag?area covered by each size bag?

2. The length of one base of a 2. The length of one base of a trapezoid is 6 cm greater than the trapezoid is 6 cm greater than the length of the other base. The length of the other base. The height of the trapezoid is 11 cm height of the trapezoid is 11 cm and its area is 165 cm2. What are and its area is 165 cm2. What are the lengths of the bases? the lengths of the bases?

Hint: the area of a trapezoid is Hint: the area of a trapezoid is

A 1

2h b1 b2

3. Twice the sum of two 3. Twice the sum of two consecutive integers is 246. Let consecutive integers is 246. Let nn = the smaller integer. = the smaller integer.

4. Each of the two congruent sides 4. Each of the two congruent sides of an isosceles triangle is 10 cm of an isosceles triangle is 10 cm shorter than its base, and the shorter than its base, and the perimeter of the triangle is 205 perimeter of the triangle is 205 cm. Let cm. Let xx = the length of the = the length of the base. base.

2.4 Solving 2.4 Solving InequalitiesInequalities

NotationNotation

The symbol The symbol is used to represent is used to represent “less than”“less than”

The symbol The symbol is used to represent is used to represent “less than or equal to” “less than or equal to”

The symbol The symbol is used to represent is used to represent “greater than”“greater than”

The symbol The symbol is used to represent is used to represent “greater than or equal to”“greater than or equal to”

Properties of Properties of InequalitiesInequalities

1. If 1. If a, ba, b, and , and cc are real numbers, and if are real numbers, and if and , and , then then

2.2. To solve inequalities, you can add or subtract To solve inequalities, you can add or subtract the same number to both sides of the inequality:the same number to both sides of the inequality:IfIf , then , then . .

3. To solve inequalities, you can multiply or divide 3. To solve inequalities, you can multiply or divide by the same number on both sides. by the same number on both sides. However,However, if if you multiply or divide both sides by a negative you multiply or divide both sides by a negative number, you number, you the inequality. the inequality.Example: Multiply both sides of Example: Multiply both sides of by -1 and by -1 and see what happens!see what happens!

a bb c a c

a b a c b c

flip62

Graphing Inequalities Graphing Inequalities on a Number Lineon a Number Line

1.1. Solve the inequality. Keep the variable on Solve the inequality. Keep the variable on the the leftleft side of the equation. side of the equation.

2.2. If the inequality is < or >, use an If the inequality is < or >, use an circle. If the inequality is or circle. If the inequality is or use a use a

circle. circle. 3. Shade the number line in the direction 3. Shade the number line in the direction

that makes the inequality true. If you keep that makes the inequality true. If you keep the variable on the left, you will shade in the variable on the left, you will shade in the direction the inequality points.the direction the inequality points.

open

closed

Solve the inequality Solve the inequality and graph its solution and graph its solution

set set

1. 1. 2a 113

Solve the inequality Solve the inequality and graph its solution and graph its solution

set set

2. 2. 1

3n 5 11

3

Solve the inequality Solve the inequality and graph its solution and graph its solution

set set

3. 3. 6 1 2h 7h 9

Solve the inequality Solve the inequality and graph its solution and graph its solution

set set

4. 4. 2 4 5v 3

13 5v 6

21

2.5 Compound 2.5 Compound SentencesSentences

A A sentence has sentence has either an either an or an or an . .

If the joiner is an If the joiner is an that means that means that both sentences need to be true.that both sentences need to be true.

If the joiner is an If the joiner is an that means that means that only one sentence or the other that only one sentence or the other needs to be true.needs to be true.

compound

and or

and

or

For example, For example, is the is the same this as saying same this as saying

andand

x3

73 x

7x

Graphically,Graphically, also written as also written as

andand

x3 3x

73

7x

73

So, the solution So, the solution would look would look like like

73 x

73

An An oror statement, on the other hand statement, on the other hand would look different since only ONE would look different since only ONE of the inequalities has to be true.of the inequalities has to be true.

For example, For example, or or

Would 7 be a solution? Would 7 be a solution?

Would 0 be a solution? Would 0 be a solution?

Would 4 be a solution?Would 4 be a solution?

6x 2x

yes

yes

no

Graphically,Graphically,

or or

6x

62

2x

62

So, the solution So, the solution or or would look like would look like

6x 2x

62

When solving compound sentences When solving compound sentences where the variable is in the middle of where the variable is in the middle of two inequalities, set it up like an two inequalities, set it up like an andand problem to solve. Combine your problem to solve. Combine your inequalities into one statement at the inequalities into one statement at the end.end.

When solving a compound sentence When solving a compound sentence that is an that is an oror problem, solve each problem, solve each inequality and then graph them both.inequality and then graph them both.

Solve the open sentence Solve the open sentence and graph its solution and graph its solution set. set.

1. 1. 7123 d

Solve the open sentence Solve the open sentence and graph its solution and graph its solution set. set.

2. 2. 102

196 t

Solve the open sentence Solve the open sentence and graph its solution and graph its solution set. set.

3. 3. 10

1112

2

37

5

4 yyy

Solve the open sentence Solve the open sentence and graph its solution and graph its solution set. set.

4.4. or or 5

3

5

47

d4

21 d

Solve the open sentence Solve the open sentence and graph its solution and graph its solution set. set.

5.5. or or 53411 v 3275 vv