Lesson 74: Solving Absolute-Value Equations

New Concept

The absolute value of a number $n$ is the distance from $n$ to 0 on a number line.

What’s next

Next, you’ll apply this definition to solve equations where the variable's value can be one of two distinct possibilities.

Property

An equation that has one or more absolute-value expressions is called an absolute-value equation. The absolute value of a number $n$ is the distance from $n$ to 0 on a number line.

Explanation

Think of absolute value as a 'distance detector'—it only measures how far a number is from zero, ignoring whether it's positive or negative. So, an equation like $|x| = 5$ simply asks, 'Which numbers are exactly 5 steps away from zero?' This always leads to two possibilities, one on each side of zero.

Examples

• If $|x| = 7$, it means $x$ is 7 units from 0, so $x = 7$ or $x = -7$.
• The expression $|y+2| = 6$ means the entire quantity $(y+2)$ must be 6 units away from 0.
• The solution set for $|z| = 13$ is $\{13, -13\}$ because both numbers are 13 units from 0.

An absolute value equation often hides two different solutions in plain sight. Let's practice solving one by splitting it into two separate cases.

Example Problem

Solve the equation $|x+5| = 12$.

Step-by-Step

1. Using the definition of absolute value, we can write $|x+5| = 12$ as two separate equations.
2. Case 1: The expression inside the absolute value is positive.

3. Subtract 5 from both sides to solve for $x$.

4. Case 2: The expression inside the absolute value is negative.

5. Subtract 5 from both sides to solve for $x$.

6. The solution set contains both values we found.

Property

To solve an absolute-value equation like $|A| = b$ where $b > 0$, you must create and solve two separate linear equations: $A = b$ or $A = -b$.

Explanation

Solving an absolute value equation is like exploring a forked road. Since the expression inside could be positive or negative to start with, you must follow both paths to find all possible answers. Just remember to drop the absolute value bars once you split the equation into its two separate, simpler versions to solve.

Examples

• To solve $|x+3| = 16$, create two paths: $x+3 = 16$ (which gives $x=13$) and $x+3 = -16$ (which gives $x=-19$).
• Solving $|y-5| = 11$ gives you $y-5=11$ (so $y=16$) and $y-5=-11$ (so $y=-6$).
• The equation $|2k| = 10$ splits into two possibilities: $2k=10$ and $2k=-10$, so the solutions are $k=5$ or $k=-5$.

Property

In some equations, it is necessary to first isolate the absolute-value term before splitting it into two equations.

Explanation

Before you can split the equation into its two positive and negative paths, you have to get the absolute value expression completely by itself! Think of it like unwrapping a gift. You need to get rid of any numbers multiplied or added outside the bars first. Only then can you properly see what's inside and solve.

Examples

• For $5|x| = 20$, first divide both sides by 5 to get $|x| = 4$. Then you can solve for $x=4$ or $x=-4$.
• In $4|x-2| = 80$, divide by 4 to get $|x-2| = 20$. Now you are ready to split and solve the two cases.
• To solve $|x-6|-2 = -2$, first add 2 to both sides to isolate the absolute value, which gives you $|x-6|=0$.

Sometimes the absolute value part isn't by itself. Let's see how to clear away the clutter first before solving.

Example Problem

Solve the equation $3|x-4| = 45$.

Step-by-Step

1. To begin, we must isolate the absolute value expression.
2. Use the Division Property of Equality to divide both sides by 3.

3. Simplify the equation.

4. Now that the absolute value is isolated, write $|x-4|=15$ as two equations.

5. Solve the first equation by adding 4 to both sides.

6. Solve the second equation by adding 4 to both sides.

7. The final solution is the set of both values.