Solving Compound Solving Compound InequalitiesInequalities

1. Solve compound inequalities containing the word and then graph the solution.

2. Solve compound inequalities containing the word or then graph the solution.

Inequalities containing Inequalities containing ANDAND Two inequalities joined by the word AND. True only if both of the inequalities are

true. The graph of a compound inequality

containing the word AND is the INTERSECTION of the graphs of the two inequalities.

The solution MUST be a solution of BOTH inequalities.

The INTERSECTION can be found by determining where the graphs OVERLAP!

Graph an IntersectionGraph an Intersection Graph the solution set of x < 3 and x ≥ -2.

--33

-2-2 -1-1 00 11 22 33 44

--33

-2-2 -1-1 00 11 22 33 44

--33

-2-2 -1-1 00 11 22 33 44

Graph first Graph first SolutionSolution

Graph Second Graph Second SolutionSolution

Graph the Graph the Intersection of Intersection of the two graphsthe two graphs

Solve and Graph an Solve and Graph an IntersectionIntersection -5 < x – 4 < 2 Write the inequality as an AND

statement -5 < x – 4 and x – 4 < 2 Solve each inequality -5 < x – 4 x – 4 < 2 +4 +4 +4 +4 -1 < x x < 6 On the Next Slide, you will see the

graphs.

The Graphs

x > -1 (The 1st solution)

x < 6 (The 2nd solution)

-1 < x < 6 (The INTERSECTION)

00--11

--22

--33

11 22 33 44 55 66 77

00--11

--22

--33

11 22 33 44 55 66 77

00--11

--22

--33

11 22 33 44 55 66 77

Use open circles because of the inequality symbols.

This is the INTERSECTION of the two graphs!

Inequalities containing Inequalities containing OROR These compound inequalities are true

if one or more of the inequalities are true.

The graph of a compound inequality containing the word OR is the UNION of the graphs of the two inequalities.

The solution of the compound inequality is a solution of either inequality.

The UNION can be found by graphing both inequalities.

Write and Graph an Write and Graph an InequalityInequality An airplane is experiencing turbulence while flying

at 30,000 feet. The control tower tells the pilot that he should increase his altitude to at least 35,000 feet or decrease his altitude to no more than 25,000 feet to avoid the turbulence.

WORDS The pilot needs to get the plane at least 35,000

feet or no more than 25,000 feet. VARIABLES Let a represent the plane’s altitude INEQUALITY a ≥ 35,000 or a ≤ 25,000 Graph to follow on next Slide.

GRAPH of AIRPLANE GRAPH of AIRPLANE PROBLEMPROBLEM a ≥ 35,000 or a ≤ 25,000

30,0030,0000

25,0025,0000

35,00035,000

30,0030,0000

25,0025,0000

35,00035,000

30,0030,0000

25,0025,0000

35,00035,000

First InequalityFirst Inequality

Second Second InequalityInequality

UnionUnion

Solve and Graph a UnionSolve and Graph a Union

-3x + 4 < 19 or 7x – 3 > 18 -4 -4 +3 +3 -3x < 15 or 7x > 21 -3 -3 7 7 x > -5 or x > 3 Graph on the Next Slide.

The GraphThe Graph x > -5 or x > 3

-5-5 00 55

-5-5 00 55

-5-5 00 55

1st Inequality

2nd Inequality

Union

Notice the graph of x > -5 contains all the points in x > 3. So the union is the graph of x > -5.