5.1 Addition and Subtraction Problems of Inequality

Objective: •To solve and graph the solution set of an inequality by using the Addition or Subtraction Property of Inequality

Frameworks: 10.P.1, 10.P.7

Quarter 2 Reality Check

1. “Math class is tough.” Barbie, 1992

2. Life is not fair.3. Everyone in this class needs to

A. Work hard. B. Not disrupt others who are trying to succeed.

How do you read . . .

a < b a is less than ba > b a is greater than b

Inequality

The open sentence x < -2 is an example of an inequality

An inequality contains at least one variable and consists of 2 expressions with an inequality symbol such as <, >, or ≠ between them.

Solving an Inequality

To solve an inequality means to find a solution set.

What is the solution set of x < -2?

On a number line:open circle meansnot including this point

Solving an Inequality

How would we graph the solution of x > 1?

Solving an Inequality

The Addition and Subtraction Properties of Equality allow you to add or subtract the same number from each side of an equation to obtain an equivalent equation.

x – 4 = 3 x + 2 = 5

Do inequalities work the same way?

Solving an Inequality

2 < 6 +5 +5 7 < 11 TRUE

Solving an Inequality

2 < 6 -1 -1 1 < 5 TRUE

Equivalent Inequalities

Open inequalities with the same solution set are called equivalent inequalities.

Addition Property of Inequality

For all real numbers a, b, and c, if a < b, then a + c < b + c, and if a > b, then a + c > b + c

In other words, adding the same number to each side of an equality produces an equivalent equality.

Subtraction Property of Inequality

For all real numbers a, b, and c, if a < b, then a - c < b - c, and if a > b, then a - c > b - c

In other words, subtracting the same number from each side of an equality produces an equivalent equality.

Solve x – 8 > -11 & Graph

Solve & Graph 7 < 5 – (½ – x)

After Mary paid $8.36 for a snack she had less than $2.50 left. How much money did she have originally?

After Bill paid $7.21 at the movies, he had less than $1.75 left. How much money did he have originally?

5.2 Multiplication & Division Problems of Inequality

Objective: •To solve and graph the solution set of an inequality by using the Multiplication or Division Property of Inequality

Frameworks: 10.P.1, 10.P.7

Solving an Inequality

The Multiplication and Division Properties of Equality allow you to add or subtract the same number from each side of an equation to obtain an equivalent equation.

x / 4 = 2 x * 3 = 21

Do inequalities work the same way?

Solving an Inequality

3 < 4 *5 *5 15 < 20 TRUE

Solving an Inequality

-4 > -20 /2 /2 -2 > -10 TRUE

Solving an Inequality

-5 < -3 *-1 *-1 5 < 3 FALSE

Solving an Inequality

18 > -6 /-3 /-3 -6 > 2 FALSE

Notice:

Multiplying or Dividing each side of a true equality by a negative number produces a false inequality

Multiplication Property of Inequality, Part 1

For all real numbers a, b, and c, if a < b and c > 0, then ac < bc, and if a > b and c > 0, then ac > bc

That is, multiplying each side of an inequality by the same positive number produces an equivalent inequality.

Multiplication Property of Inequality, Part 2

For all real numbers a, b, and c, if a < b and c < 0, then ac > bc, and if a > b and c < 0, then ac < bc

That is, multiplying each side of an inequality by the same negative number and reversing the order of the inequality produces an equivalent inequality.

Division Property of Inequality, Part 1

For all real numbers a, b, and c, if a < b and c > 0, then a/c < b/c, and if a > b and c > 0, then a/c > b/c

That is, dividing each side of an inequality by the same positive number produces an equivalent inequality.

Division Property of Inequality, Part 2

For all real numbers a, b, and c, if a < b and c < 0, then ac > b/c, and if a > b and c < 0, then ac < b/c

That is, dividing each side of an inequality by the same negative number and reversing the order of the inequality produces an equivalent inequality.

Solve:

7x < -56

Divide each side by 7

x < -8Graph:

Solve:

-⅔ x > 16

Multiply each side by the reciprocal of -⅔ Because we multiplied by a negative,

change the > to a <x < -24Graph:

Solve:

-4 < - 2x

If Jill sells more than $100 worth of peanut brittle, she will win a radio. Each box of peanut brittle sells for $2.75. How many boxes must she sell to win the radio?

2.75p > 100

p > 100/2.75p > 36.3636Can she sell 36.36 boxes?Jill must sell 37 boxes.

-3x + 6 < -5

5 – 4x < 2x - 7

-3/2 x + 4 > 7

-2(2x + 1) + 5x < x + 5

Turn to p. 168

Do 1 -14Turn to p. 169Do 16-19Turn to p. 173Do 1-9Turn to p. 174Do 27-30