Section 4.2

Solving Linear Inequalities Using the Multiplication-Division Principle

4.2 Lecture Guide: Solving Linear Inequalities Using the Multiplication-Division Principle

Objective 1: Solve linear inequalities using the multiplication-division principle for inequalities.

Verbally

Order Preserving: If ____________ sides of an inequality are multiplied or ____________ by a positive number, the result is an inequality that has the same ____________ as the original inequality.

Algebraically

If a, b, and c, are real numbers then , then is __________to .

Numerical Examples

is equivalent

to

And to .

Multiplication-Division Principle for Inequalities:

32x

2 2 32x

6x

0c a b

ac bc

Verbally

Order Reversing: If ____________ sides of an inequality are multiplied or divided by a negative number and the order of inequality is ____________, the result is an inequality that has the ____________ solution as the original inequality.

Algebraically

If a, b, and c, are real numbers then , then is equivalent to .

Numerical Examples

is equivalent

to

and to .

Multiplication-Division Principle for Inequalities:

0c a b

ac bc

53x

3 3 53x

15x

Use the multiplication-division principle of equality to solve each inequality.

2.1. 8 72x 12a

Use the multiplication-division principle of equality to solve each inequality.

4.3. 45x 7 5 23x

Use the multiplication-division principle of equality to solve each inequality.

6.5. 2 3 1 2 22x x 3 4 5 5 2 1x x

Use the multiplication-division principle of equality to solve each inequality.

7.2

85

x

Use the multiplication-division principle of equality to solve each inequality.

8. 53 2x x

Use the multiplication-division principle of equality to solve each inequality.

9. 3 4 6 2x x

Use the multiplication-division principle of equality to solve each inequality.

10. 2 3 5 2 3 15 2x x x x

5 −9 116 −6 97 −3 78 0 59 3 310 6 111 9 −1

11. Use the table to solve each equation or inequality.

(a)

(b)

(c)

x 1y 2y1 2y y

1 2y y

1 2y y

12. Use the graph to solve each equation or inequality.

2y

(a)

(b)

(c)

1 2y y

1 2y y

1 2y y1y

13. A high school band is having a fund raiser to raise money for a trip. The cost of renting a snow-cone machine for the fundraiser includes a fixed cost of $84 plus a variable cost of $0.30 per snow-cone. Snow-cones can be sold for $1.50 each.

(a) Write an equation for the cost of renting the machineand selling x snow-cones.

(b) Write an equation for the revenue generated byselling x snow-cones.

1y

2y

13. A high school band is having a fund raiser to raise money for a trip. The cost of renting a snow-cone machine for the fundraiser includes a fixed cost of $84 plus a variable cost of $0.30 per snow-cone. Snow-cones can be sold for $1.50 each.

(c) Determine the values of x for which .

(d) Interpret the meaning of the answer to part (c).

1 2y y