Systems of Equations
Adding and Subtracting Rational Expressions with Different Denominators
Graphing Linear Equations
Raising an Exponential Expression to a Power
Horizontal Line Test
Quadratic Equations
Mixed Numbers and Improper Fractions
Solving Quadratic Equations by Completing the Square
Solving Exponential Equations
Adding and Subtracting Polynomials
Factorizing simple expressions
Identifying Prime and Composite Numbers
Solving Linear Systems of Equations by Graphing
Complex Conjugates
Graphing Compound Inequalities
Simplified Form of a Square Root
Solving Quadratic Equations Using the Square Root Property
Multiplication Property of Radicals
Determining if a Function has an Inverse
Scientific Notation
Degree of a Polynomial
Factoring Polynomials by Grouping
Solving Linear Systems of Equations
Exponential Functions
Factoring Trinomials by Grouping
The Slope of a Line
Simplifying Complex Fractions That Contain Addition or Subtraction
Solving Absolute Value Equations
Solving Right Triangles
Solving Rational Inequalities with a Sign Graph
Domain and Range of a Function
Multiplying Polynomials
Slope of a Line
Multiplying Rational Expressions
Percent of Change
Equations Involving Fractions or Decimals
Simplifying Expressions Containing only Monomials
Solving Inequalities
Quadratic Equations with Imaginary Solutions
Reducing Fractions to Lowest Terms
Prime and Composite Numbers
Dividing with Exponents
Dividing Rational Expressions
Equivalent Fractions
Graphing Quadratic Functions
Linear Equations and Inequalities in One Variable
Notes on the Difference of 2 Squares
Solving Absolute Value Inequalities
Solving Quadratic Equations
Factoring Polynomials Completely
Using Slopes to Graph Lines
Fractions, Decimals and Percents
Solving Systems of Equations by Substitution
Quotient Rule for Radicals
Prime Polynomials
Solving Nonlinear Equations by Substitution
Simplifying Radical Expressions Containing One Term
Factoring a Sum or Difference of Two Cubes
Finding the Least Common Denominator of Rational Expressions
Multiplying Rational Expressions
Expansion of a Product of Binomials
Solving Equations
Exponential Growth
Factoring by Grouping
Solving One-Step Equations Using Models
Solving Quadratic Equations by Factoring
Adding and Subtracting Polynomials
Rationalizing the Denominator
Rounding Off
The Distributive Property
What is a Quadratic Equation
Laws of Exponents and Multiplying Monomials
The Slope of a Line
Factoring Trinomials by Grouping
Multiplying and Dividing Rational Expressions
Solving Linear Inequalities
Multiplication Property of Exponents
Multiplying and Dividing Fractions 3
Dividing Monomials
Multiplying Polynomials
Adding and Subtracting Functions
Dividing Polynomials
Absolute Value and Distance
Multiplication and Division with Mixed Numbers
Factoring a Polynomial by Finding the GCF
Adding and Subtracting Polynomials
The Rectangular Coordinate System
Polar Form of a Complex Number
Exponents and Order of Operations
Graphing Horizontal and Vertical Lines
Invariants Under Rotation
The Addition Method
Solving Linear Inequalities in One Variable
The Pythagorean Theorem
Try the Free Math Solver or Scroll down to Tutorials!












Please use this form if you would like
to have this math solver on your website,
free of charge.

Absolute Value and Distance

If a is a real number, the absolute value of a is

The absolute value of a number cannot be negative. For example, let a = -4. Then, -4 < 0, because you have

| a | = | -4 | = -(-4) = 4.

Remember that the symbol -a does not necessarily mean that is -a negative.

Operations with Absolute Value

Let a and b be real numbers and let n be a positive integer.


Properties of Inequalities and Absolute Value

Let a and b be real numbers and let k be a positive real number.

1. -| a | ≤ a ≤ | a |

2. | a | ≤ k if and only if -k ≤ a ≤ k.

3. k ≤ | a | if and only if k ≤ a or a ≤ -k.

4. Triangle Inequality: | a + b | ≤ | a | + | b |

Properties 2 and 3 are also true if ≤ is replaced by <.


Example 1

Solving an Absolute Value Inequality

Solve | x - 3| ≤ 2.


Using the second property of inequalities and absolute value, you can rewrite the original inequality as a double inequality.

-2 x - 3 ≤ 2 Write as double inequality.
-2 + 3 x - 3 + 3 ≤ 2 + 3 Add 3.
1 x ≤ 5 Simplify.

The solution set is [1, 5], as shown in the figure below.


Example 2

A Two-Interval Solution Set

Solve 3 < | x + 2 |


Using the third property of inequalities and absolute value, you can rewrite the original inequality as two linear inequalities.

3 < x + 2 or x + 2 < -3
1 < x or x < -5

The solution set is the union of the disjoint intervals (-∞, -5) and (1,) as shown in the figure below.

Examples 1 and 2 illustrate the general results shown in the figure below.

Note that if d > 0, the solution set for the inequality | x - a | ≤ d is a single interval, whereas the solution set for the inequality | x - a | ≥ d is the union of two disjoint intervals.

The distance between two points a and b on the real line is given by d = | a - b | = | b - a |

The directed distance from a to b is b - a and the directed distance from b to a is as shown in the figure below.


Example 3

Distance on the Real Line

a. The distance between -3 and 4 is or

| 4 - (-3)| = | 7 | = 7 or | -3 - 4 | = | -7 | = 7.

(See the figure below.)

b. The directed distance from -3 to 4 is 4 - (-3) = 7.

c. The directed distance from 4 to -3 is -3 - 4 = -7.

The midpoint of an interval with endpoints a and b is the average value of a and b. That is,

To show that this is the midpoint, you need only show that (a + b)/2 is equidistant from a and b.

All Right Reserved. Copyright 2005-2019