Lesson 101: Solving Multi-Step Absolute-Value Inequalities

New Concept

The expression $|c - 29.75|$ represents the difference between the actual circumference and 29.75 inches.

What’s next

Next, you'll use this idea to solve multi-step inequalities and apply it to problems like defining acceptable product specifications.

Property

First, isolate the absolute value expression. Then, rewrite it as a compound inequality. Use AND for "less than" ($|x| < c \rightarrow -c < x < c$) and OR for "greater than" ($|x| > c \rightarrow x < -c$ or $x > c$).

Explanation

Think of it like unwrapping a gift! You have to get the absolute value expression all by itself on one side before you can see the two possible solutions hidden inside. It's the most important first step, so don't rush past it. After isolating, you split the problem into two separate paths.

Examples

• Solve $3|x| + 5 < 14$. First, isolate $|x|$ to get $|x| < 3$. Then, unwrap it: $-3 < x < 3$.
• Solve $|x+2| - 4 > 1$. First, isolate the absolute value to get $|x+2| > 5$. Then, split it: $x+2 < -5$ or $x+2 > 5$, which simplifies to $x < -7$ or $x > 3$.
• Solve $|\frac{x}{2} - 1| + 6 \leq 10$. Isolate to get $|\frac{x}{2} - 1| \leq 4$. This becomes $-4 \leq \frac{x}{2} - 1 \leq 4$. Solving gives $-6 \leq x \leq 10$.

Property

When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol.

Explanation

Dividing by a negative is like entering a "mirror world" where everything is backward! Greater than becomes less than, and less than becomes greater than. Forgetting to flip the sign is a super common mistake, so always double-check when you see that negative sign in a multiplication or division step.

Examples

• Solve $-5|x| + 10 \geq -15$. Subtract 10 to get $-5|x| \geq -25$. Divide by $-5$ and FLIP the sign: $|x| \leq 5$. The final answer is $-5 \leq x \leq 5$.
• Solve $-2|x-1| > -8$. Divide by $-2$ and FLIP the sign from $>$ to $<$: $|x-1| < 4$. Then solve: $-4 < x-1 < 4$, which gives $-3 < x < 5$.

Property

A value $c$ that can vary no more than some amount $v$ from a standard value $s$ is modeled by the inequality $|c - s| \leq v$.

Explanation

This sounds tricky, but it's just about setting a tolerance or a margin of error. The formula $|c - 29.75| \leq 0.25$ means the basketball's circumference ($c$) can't be "off" from the ideal 29.75 inches by more than 0.25 inches. It helps keep the game fair for everyone who plays!

Examples

• NCAA rules state a ball's circumference $c$ must vary no more than 0.25 inches from 29.75 inches. This is modeled by $|c - 29.75| \leq 0.25$, so the acceptable range is $29.5 \leq c \leq 30$ inches.
• A ball's weight $w$ must vary no more than 1 ounce from 21 ounces. We write this as $|w - 21| \leq 1$. This means the acceptable weight is between 20 and 22 ounces, or $20 \leq w \leq 22$.