Solve: $|2 x-4| \leq 8$
Give your answer as an interval. Enter DNE if the inequality does not exist.
Answer (1)
Rewrite the absolute value inequality as a compound inequality: $-8
\leq 2x - 4 \leq 8$.
Add 4 to all parts: − 4 ≤ 2 x ≤ 12 .
Divide all parts by 2: − 2 ≤ x ≤ 6 .
Express the solution as an interval: [ − 2 , 6 ] .
Explanation
Understanding the Problem We are given the absolute value inequality $|2x - 4|
\leq 8 an d a s k e d t oso l v e f or x$ and express the solution as an interval.
Rewriting the Inequality Recall that an absolute value inequality of the form $|a|
\leq b c anb ere w r i tt e na s a co m p o u n d in e q u a l i t y : -b
\leq a \leq b$. Applying this to our problem, we can rewrite the given inequality as:
− 8 ≤ 2 x − 4 ≤ 8
Isolating x To solve for x , we need to isolate x in the middle of the inequality. First, we add 4 to all parts of the inequality:
− 8 + 4 ≤ 2 x − 4 + 4 ≤ 8 + 4
Simplifying the Inequality This simplifies to:
− 4 ≤ 2 x ≤ 12
Dividing by 2 Next, we divide all parts of the inequality by 2:
2 − 4 ≤ 2 2 x ≤ 2 12
Final Simplification This simplifies to:
− 2 ≤ x ≤ 6
Expressing the Solution as an Interval Therefore, the solution to the inequality is all x such that − 2 ≤ x ≤ 6 . We can express this solution as an interval: [ − 2 , 6 ] .
Examples
Absolute value inequalities are useful in many real-world situations. For example, suppose a machine is designed to fill bags with 500 grams of sugar, but the actual amount can vary by up to 5 grams. This means the amount of sugar, x , must satisfy the inequality ∣ x − 500∣ ≤ 5 . Solving this inequality tells us the range of possible sugar amounts the bag can contain. In this case, it would be between 495 and 505 grams.
