Algebra Tutorials!
Friday 23rd of March
 Home Exponential Decay Negative Exponents Multiplying and Dividing Fractions 4 Evaluating Expressions Involving Fractions The Cartesian Coordinate System Adding and Subtracting Fractions with Like Denominators Solving Absolute Value Inequalities Multiplying Special Polynomials FOIL Method Inequalities Solving Systems of Equations by Graphing Graphing Compound Inequalities Solving Quadratic Equations by Completing the Square Addition Property of Equality Square Roots Adding and Subtracting Fractions The Distance Formula Graphing Logarithmic Functions Fractions Dividing Mixed Numbers Evaluating Polynomials Power of a Product Property of Exponents Terminology of Algebraic Expressions Adding and Subtracting Rational Expressions with Identical Denominators Solving Exponential Equations Factoring The Difference of 2 Squares Changing Fractions to Decimals Solving Linear Equations Using Patterns to Multiply Two Binomials Completing the Square Roots of Complex Numbers Methods for Solving Quadratic Equations Conics in Standard Form Solving Quadratic Equations by Using the Quadratic Formula Simplifying Fractions 2 Exponential Notation Exponential Growth The Cartesian Plane Graphing Linear Functions The Slope of a Line Finding Cube Roots of Large Numbers Rotating Axes Common Mistakes With Percents Solving an Equation That Contains a Square Root Rational Equations Properties of Common Logs Composition of Functions Using Percent Equations Solving Inequalities Properties of Exponents Graphing Quadratic Functions Factoring a Polynomial by Finding the GCF The Rectangular Coordinate System Adding and Subtracting Fractions Multiplying and Dividing Rational Expressions Improper Fractions and Mixed Numbers Properties of Exponents Complex Solutions of Quadratic Equations Solving Nonlinear Equations by Factoring Solving Quadratic Equations by Factoring Least Common Multiples http: Solving Exponential Equations Solving Linear Equations Multiplication Property of Equality Multiplying Mixed Numbers Multiplying Fractions Reducing a Rational Expression to Lowest Terms Literal Numbers Factoring Trinomials Logarithmic Functions Adding Fractions with Unlike Denominators Simplifying Square Roots Adding Fractions Equations Quadratic in Form Dividing Rational Expressions Slopes of Parallel Lines Simplifying Cube Roots That Contain Variables Functions and Graphs Complex Numbers Multiplying and Dividing Fractions 1 Composition of Functions Intercepts of a Line Powers http: Multiplying Two Numbers with the same Tens Digit and whose Ones Digits add up to 10 Factoring Trinomials Exponents and Polynomials Decimals and their Equivalent Fractions Negative Integer Exponents Adding and Subtracting Mixed Numbers Solving Quadratic Equations Theorem of Pythagoras Equations 1 Subtracting Fractions Solving Quadratic Equations by Graphing Evaluating Polynomials Slope Angles and Degree Measure
Try the Free Math Solver or Scroll down to Tutorials!

 Depdendent Variable

 Number of equations to solve: 23456789
 Equ. #1:
 Equ. #2:

 Equ. #3:

 Equ. #4:

 Equ. #5:

 Equ. #6:

 Equ. #7:

 Equ. #8:

 Equ. #9:

 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

 Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:

# Solving Absolute Value Inequalities

## Solving an Absolute Value Inequality of the Form | x| > a

Principle

Absolute Value Inequalities of the Form | x| > a and | x| ≥ a

Let a represent a positive real number.

 â€¢ If |x| > a, then x < -a or x > a. â€¢ If |x| ≥ a, then x ≤ -a or x ≥ a.

â€¢ If |x| > 0, then the solution is all real numbers, except 0.

â€¢ If |x| 0, then the solution is all real numbers.

â€¢ If |x| > -a, then the solution is all real numbers.

â€¢ If |x| ≥ -a, then the solution is all real numbers.

Note:

The absolute value of a number or expression is always greater than a negative number.

Next, letâ€™s solve some absolute value inequalities of the form |x| > a and x| a.

Example 1

Solve: 3|5x| ≥ 60.

 Solution Step 1 Isolate the absolute value. Divide both sides by 3. Step 2 Make the substitution w = 5x. Step 3 Use the Absolute Value Principle to solve for w. Step 4 Replace w with 5x. Step 5 Solve for x. Divide each side by 5. 3|5x| ≥ 60  |5x| ≥ 20 |w| ≥ 20 w ≤ -20 or w ≥ 20 5x ≤ -20 or 5x ≥ 20   x ≤ -4 or x ≥ 4

There are infinitely many solutions. It is a good idea to check one number from each part of the solution. Letâ€™s check x = -6 and x = 7.

 Check x = -6 Check x = 7 Is Is Is Is 3|5x| ≥ 3|5(-6)| ≥ 3|-30| ≥ 3 Â· 30 ≥ 90 ≥ 6060 ? 60 ? 60 ? 60 ? Yes Is Is Is Is 3|5x| ≥ 3|5(7)| ≥ 3|35| ≥ 3 Â· 35 ≥ 105 ≥ 6060 ? 60 ? 60 ? 60 ? Yes
So, the solution is x -4 or x 4.

Note:

You may be tempted to write x -4 or x 4 as -4 x 4.

However, this implies that -4 is greater than 4, which is false.

We cannot write x -4 or x 4 as a single compound inequality.