Previously you studied four types of simple inequalities. In this lesson you

will study compound inequalities. A compound inequality consists of two

inequalities connected by and or or.

Solving Compound Inequalities

Write an inequality that represents the set of numbers and graph the inequality.

All real numbers that are greater than zero and less than or equal to 4.

SOLUTION 0 < x 4

This inequality is also written as 0 < x and x 4.

40 1 2 3 5–1

Previously you studied four types of simple inequalities. In this lesson you

will study compound inequalities. A compound inequality consists of two

inequalities connected by and or or.

Solving Compound Inequalities

Write an inequality that represents the set of numbers and graph the inequality.

All real numbers that are greater than zero and less than or equal to 4.

All real numbers that are less than –1 or greater than 2.

SOLUTION 0 < x 4

This inequality is also written as 0 < x and x 4.

x < –1 or x > 2

40 1 2 3 5–1

3–1 0 1 2 4–2SOLUTION

Write an inequality that describes the elevations of the regions of Mount Rainier.

Compound Inequalities in Real Life

Timber region below 6000 ft

SOLUTION

Let y represent the approximate elevation (in feet).

Timber region:

2000 y < 6000

Alpine meadow region

Timber region

Glacier and permanent snow field region

14,140 ft

7500 ft6000 ft

2000 ft

0 ft

Write an inequality that describes the elevations of the regions of Mount Rainier.

Compound Inequalities in Real Life

Timber region below 6000 ft

SOLUTION

Let y represent the approximate elevation (in feet).

Timber region:

2000 y < 6000

Alpine meadow region

Timber region

Glacier and permanent snow field region

14,140 ft

7500 ft6000 ft

2000 ft

0 ft

Alpine meadow region below 7500 ft

Alpine meadow region:

6000 y < 7500

Write an inequality that describes the elevations of the regions of Mount Rainier.

Compound Inequalities in Real Life

Timber region below 6000 ft

SOLUTION

Let y represent the approximate elevation (in feet).

Timber region:

2000 y < 6000

Alpine meadow region

Timber region

Glacier and permanent snow field region

14,140 ft

7500 ft6000 ft

2000 ft

0 ft

Alpine meadow region below 7500 ft

Alpine meadow region:

6000 y < 7500

Glacier and permanent snow field region

Glacier and permanent snow field region:

7500 y  14,410

Solve –2 3x – 8 10. Graph the solution.

Solving a Compound Inequality with And

SOLUTION

Isolate the variable x between the two inequality symbols.

–2 3 x – 8 10 Write original inequality.

6 3 x 18 Add 8 to each expression.

2 x 6 Divide each expression by 3.

The solution is all real numbers that are greater than or equal to 2and less than or equal to 6.

40 1 2 3 65 7

Solve 3x + 1 < 4 or 2x – 5 > 7. Graph the solution.

Solving a Compound Inequality with Or

SOLUTION

A solution of this inequality is a solution of either of its simple parts.You can solve each part separately.

The solution is all real numbers that are less than 1 or greater than 6.

3x + 1 < 4 2x – 5 > 7or

3x < 3 2x > 12or

x < 1 x > 6or

40 1 2 3 65–1 7

Solve –2 < –2 – x < 1. Graph the solution.

Reversing Both Inequality Symbols

SOLUTION

Isolate the variable x between the two inequality signs.

–2 < –2 – x < 1

0 < –x < 3

0 > x > –3

To match the order of numbers on a number line, this compound is usually written as –3 < x < 0. The solution is all real numbers that are greater than–3 and less than 0.

Write original inequality.

Add 2 to each expression.

Multiply each expression by –1 andreverse both inequality symbols.

0– 4 –3 –2 –1 21–5 3

You have a friend Bill who lives three miles from school and another friend

Mary who lives two miles from the same school. You wish to estimate the

distance d that separates their homes.

What is the smallest value d might have?

Modeling Real-Life Problems

DRAW A DIAGRAM A good way to begin this problem is to draw a diagram with the school at the center of a circle.

ProblemSolvingStrategy

SOLUTION

Bill’s home is somewhere on the circle with radius 3 miles and center at the school.

Mary’s home is somewhere on the circle with radius 2 miles and center at the school.

You have a friend Bill who lives three miles from school and another friend

Mary who lives two miles from the same school. You wish to estimate the

distance d that separates their homes.

What is the smallest value d might have?

Modeling Real-Life Problems

If both homes are on the same line going toward school, the distance is 1 mile.

SOLUTION

You have a friend Bill who lives three miles from school and another friend

Mary who lives two miles from the same school. You wish to estimate the

distance d that separates their homes.

What is the largest value d might have?

Modeling Real-Life Problems

SOLUTION

If both homes are on the sameline but in opposite directions fromschool, the distance is 5 miles.

You have a friend Bill who lives three miles from school and another friend

Mary who lives two miles from the same school. You wish to estimate the

distance d that separates their homes.

What is the largest value d might have?

Modeling Real-Life Problems

SOLUTION

If both homes are on the sameline but in opposite directions fromschool, the distance is 5 miles.

Write an inequality that describes all the possible values that d might have.

The values of d can be described by the inequality 1 d 5.

SOLUTION