Lesson 94: Solving Multi-Step Absolute-Value Equations

New Concept

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

What’s next

Next, you’ll learn to isolate the absolute value expression and solve the two resulting linear equations to find the solutions.

Property

To solve an absolute-value equation, begin by isolating the absolute value. Then use the definition of absolute value to write the absolute-value equation as two equations. Solve each equation, and write the solution set.

Explanation

Your first mission is to get the absolute value expression all by itself using inverse operations. Once it's isolated, you crack it open into two separate equations—one for the positive outcome and one for the negative. Solve both of these simpler problems, and you've found all the secret answers! It's like one map leading to two treasures.

Examples

Solve $4|x+2| - 5 = 11$. Isolate: $4|x+2|=16 \rightarrow |x+2|=4$.
Split into two equations: $x+2=4$ or $x+2=-4$.
The final solutions are $x=2$ or $x=-6$.

This time, the absolute value holds a binomial. Don't worry, the strategy is the same. This problem illustrates our second key idea: solving for a more complex expression inside the absolute value bars.

Example Problem

Solve the equation $4|x+5| - 2 = 18$.

Property

If an isolated absolute value expression equals 0, there is only one solution. If it equals a negative number, there is no solution, written as $\emptyset$.

Explanation

Watch out for these special scenarios! If your isolated absolute value equals zero, you've hit the jackpot with just one answer. But if it equals a negative number, stop right there! It's a mathematical trap. Since absolute value represents distance, it can't be negative, so there's no solution at all. An empty treasure chest!

Examples

Solve $\frac{7|x|}{3} + 5 = 5$. Isolating gives $|x|=0$. The only solution is $x=0$.
Solve $2|x| + 12 = 4$. Isolating gives $2|x|=-8$, so $|x|=-4$. This is impossible, so the solution is no solution, $\emptyset$.

Property

Use the equation $|5x - 100| = 10$ to find the least and greatest number of items the factory can produce.

Explanation

Real-world situations like budgets often use absolute value to define a 'plus or minus' range. This equation shows a factory wanting its production cost to be 100 dollars, give or take 10 dollars. Solving the absolute value equation reveals the minimum and maximum number of items they can produce while staying within that financial wiggle room.

Examples

The equation $|5x - 100| = 10$ splits into two possibilities for the cost.
Case 1 (greatest): $5x - 100 = 10 \Rightarrow 5x=110 \Rightarrow x=22$ items.
Case 2 (least): $5x - 100 = -10 \Rightarrow 5x=90 \Rightarrow x=18$ items.