SOLVING INEQUALITIES Chapter 3. Introduction In this chapter we will extend the skills learned in the previous chapter for solving equalities to inequalities

SOLVING INEQUALITIESChapter 3

Introduction

• In this chapter we will extend the skills learned in the previous chapter for solving equalities to inequalities.

• Many of the procedures used will be the same reflecting that the properties of inequalities are very similar to those for equations.• Some key difference exists though.

• We will also learn to solve for equations and inequalities involving absolute values.

Inequalities and Their Graphs (3.1)Solution of an inequality: Any number that makes the inequality true.• When graphing solutions of inequalities:

• Open dot: The particular number is not part of the solution set.

• Closed dot: The particular number is a part of the solution set.

Solving Inequalities Using Addition and Subtraction (3.2)Equivalent inequalities: Inequalities with the same solutions.• We can add or subtract the same value to each side of an

inequality just as we did with equations to solve for an inequality.

Addition Property of InequalityFor every real number a, b, or c,If a > b, then a+c > b+c If a<b, then a+c<b+c

Subtraction Property of InequalityFor every real number a, b, or c,If a > b, then a-c > b-c If a<b, then a-c<b-c

Solving Inequalities Using Addition and Subtraction (3.2)

Sample Problem

a) Solve

b) Solve

c) Solve

Solving Inequalities Using Multiplication and Division (3.3)• We can multiply both sides of an inequality in much the

same way we did with an equation.• There is one key exception and it has to do with whether

we are multiplying by a negative or a positive number:

When multiplying by a positive number.

Do not change the direction of the inequality symbol.

When multiplying by a negative number.

Change the direction of the inequality symbol.

Solving Inequalities Using Multiplication and Division (3.3)There are two properties that deal with multiplying inequalities:

Multiplication Property of Inequality for c >0(c is a positive number)For every real number a, b, or c,If a > b, then ac > bc If a<b, then ac<bc

Multiplication Property of Inequality for c < 0(c is a negative number)For every real number a, b, or c,If a > b, then ac < bc If a<b, then ac>bc

Solving Inequalities Using Multiplication and Division (3.3)

Sample Problem

a) Solve:

b) Solve:

Solving Inequalities Using Multiplication and Division (3.3)There are two properties that deal with dividing inequalities:

Division Property of Inequality for c >0(c is a positive number)For every real number a, b, or c,If a > b, then > If a<b, then <

Division Property of Inequality for c < 0(c is a negative number)For every real number a, b, or c,If a > b, then < If a<b, then >

Solving Inequalities Using Multiplication and Division (3.3)

Sample Problem

a) Solve:

b) Solve:

Solving Multi-Step Inequalities(3.4)• Solving multi-step inequalities is the same as solving

multi-step equations. • Simplify by dealing with the addition or subtraction first by carrying

out the inverse operation.• Then simplify the multiplication/division by again by taking the

inverse operation.• Remember, simply multiplying by the inverse of the coefficient will be the

same as applying the inverse operation.

• Again the exception occurs when we multiply or divide by a negative number to simplify the inequality:• When multiplying/dividing by a negative number remember to

reverse the inequality sign.

Solving Multi-Step Inequalities(3.4)

Sample Problem

a) Solve

b) Solve


Sample Problem

a) Solve


Sample Problem

a) Solve

b) Solve

Compound Inequalities (3.5)Compound inequality: Two inequalities that are joined by the word “and” or the word “or”.

• The solution to compound inequalities such as the one above is in the overlap of the solutions of the individual parts of the compound inequality.

Example 1

Compound Inequalities (3.5)

• Solving a compound inequality containing the word “and.”• The solution involves solving all sides the same way. • What is done to one side of the compound inequality must be

done to all sides of the compound inequality.

• Again, remember when multiplying or dividing by a negative number the inequality signs must be reversed.


Sample Problem

Solve Graph the solution.


• The solution to compound inequalities such as the one above shows that there is no overlap of the solutions of the individual parts of the compound inequality.

• When solving for compound inequalities with “or,” solve for the each inequality separately.• In solving for each inequality, use the same rules for solving

inequalities learned above.

Example 2


Sample Problem

Solve Graph the solution.

Absolute Value Equations and Inequalities (3.6)

• Remember, the absolute value of a number is its distance from zero. Therefore, the absolute value can never be negative.

• In absolute value equations, the unknowns will be set to both positive and negative values..

Solving Absolute Value EquationsTo solve an equation in the form │A│= b, whereA represents a variable expression and b > 0, solve A =b and A=


Sample Problems

a) Solve

b) Solve


• Absolute value inequalities also exist.• When solving these absolute value inequalities, write the inequalities as compound inequalities.

Solving Absolute Value InequalitiesTo solve an inequality in the form │A│< b, whereA represents a variable expression and b > 0, solve

To solve an inequality in the form │A│> b, whereA represents a variable expression and b > 0, solve


Sample Problems

a) Solve

SOLVING INEQUALITIESChapter 3

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SOLVING INEQUALITIES Chapter 3. Introduction In this chapter we will extend the skills learned in the previous chapter for solving equalities to inequalities