Solving absolute value equations

Solving Absolute Value Equations

Absolute value equations are equations that involve the absolute value function. The absolute value of a number is its distance from zero on the number line, regardless of direction. This means that for any real number x, the absolute value is defined as follows:

Definition: The absolute value of x is given by

|x| = { x, if x ≥ 0
-x, if x < 0

Understanding Absolute Value Equations

When solving an absolute value equation, you must consider the definition of absolute value. For example, the equation |x| = a has two cases:

1. x = a
2. x = -a

Thus, when you encounter an absolute value equation, you will typically create two separate equations to solve.

Examples of Solving Absolute Value Equations

Example 1

Solve the equation |x - 3| = 5.

Step 1: Identify the two cases based on the definition of absolute value.

1. x - 3 = 5
2. x - 3 = -5

Step 2: Solve each case.

For the first case:

x - 3 = 5
x = 5 + 3
x = 8

For the second case:

x - 3 = -5
x = -5 + 3
x = -2

Step 3: The solutions are x = 8 and x = -2.

Example 2

Solve the equation |2x + 1| = 7.

Step 1: Set up the two cases.

1. 2x + 1 = 7
2. 2x + 1 = -7

Step 2: Solve each case.

For the first case:

2x + 1 = 7
2x = 7 - 1
2x = 6
x = 6 / 2
x = 3

For the second case:

2x + 1 = -7
2x = -7 - 1
2x = -8
x = -8 / 2
x = -4

Step 3: The solutions are x = 3 and x = -4.

Solving Equations with Absolute Value on One Side

Sometimes, you may need to solve equations where the absolute value expression is set equal to a variable or an expression. For example, consider the equation |x + 2| = y.

To solve for x, you can set up two cases based on the value of y:

1. x + 2 = y
2. x + 2 = -y

Then, solve each case for x:

x = y - 2
x = -y - 2

This gives you two solutions for x based on the value of y.

Example 3

Now, solve the equation |3x - 4| = 10.

Step 1: Set up the two cases.

1. 3x - 4 = 10
2. 3x - 4 = -10

Step 2: Solve each case.

For the first case:

3x - 4 = 10
3x = 10 + 4
3x = 14
x = 14 / 3
x = 4.67

For the second case:

3x - 4 = -10
3x = -10 + 4
3x = -6
x = -6 / 3
x = -2

Step 3: The solutions are x = 4.67 and x = -2.

Solving Absolute Value Inequalities

In addition to equations, you may also encounter absolute value inequalities. An absolute value inequality is an expression that includes an absolute value and an inequality symbol. For example, |x| < 3.

To solve an absolute value inequality, you will split it into two cases. For |x| < a, you set up the following:

1. -a < x < a

For the inequality |x| > a, you will set it up as:

1. x < -a or x > a

Let’s see an example.

Example 4

Solve the inequality |x + 1| < 4.

Step 1: Set up the two cases.

1. -4 < x + 1 < 4

Step 2: Solve the compound inequality.

-4 < x + 1 < 4
-4 - 1 < x < 4 - 1
-5 < x < 3

Step 3: The solution is -5 < x < 3.

Summary

• Absolute value equations can be solved by creating two cases based on the definition of absolute value.
• Always check your solutions in the original equation.
• For absolute value inequalities, split them into cases depending on the inequality symbol.

Check your understanding

1. Solve the equation |x + 5| = 12.
2. Solve the equation |2x - 3| = 4.
3. Solve the inequality |x - 2| > 5.
4. What are the two cases for the equation |3x + 1| = 6?