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

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