Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Solving Compound Inequalities

Objective Learn how to solve compound inequalities.

In this lesson, we are dealing with compound inequalities, expressions in which more than one inequality applies to the same variable. When inequalities are multiplied by a negative number, the inequality symbol must be reversed.


Compound Inequalities

Let's talk a little about the notation and the terminology.

A compound inequality is an expression like 3 3x + 4 < 9 or 1x 2 7. The two inequality symbols indicate that the variable is to satisfy each part. So, for example, the inequality 2 < 2x - 5 < 4 is read “two is less than 2x - 5 and 2x - 5 is less than four” or “2x - 5 is between 2 and 4”.


Solving Compound Inequalities

First, remember that solving an inequality (compound or otherwise) means describing its solution set, since there is more than one solution. Compound inequalities can be solved using the same methods that are used to solve single inequalities, the Addition and Multiplication Properties of Inequalities.


Example 1

Solve 3 2 x + 1 8.


First, subtract 1 from each part so that no constants appear in the expression that contains the variable.

3 2x + 1 < 8  
2 2x < 7 Subtract 1 from each part.
1 x < Divide each part by 2.

The solution set is

Why is this considered solving the inequality if, after all, we still have an inequality? Why is this one any better than the original inequality?

The answer is that the inequalities now apply directly to x, not to an algebraic expression involving x. This means that it is much easier to visualize the solution set, as the points to the right of and including 1 and to the left of . Solving compound inequalities means isolating x in the inequalities.


Example 2

Solve -1 3 - x 5.


First, subtract 3 from each part to remove any constants from the expression that contains x.

-1 3 - x 5  
-4 -x 2 Subtract 3 from each side.

Next, multiply each part of the inequality by -1 in order to isolate x in the middle. Since -1 is negative, reverse the inequality symbols.

4 x -2

The solution set is { x | 4 x -2 }.

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