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Math foundations
SAT absolute value: think distance, not signs
Absolute value means distance, and that one idea solves equations, inequalities, and graphs. Slide an interactive number line to see why |x − a| = b has two answers and when it has none.
SPSAT Practice Editorial7 min read
The short version
What you'll take away
- |x − a| is the distance between x and a on the number line.
- |x − a| = b has two solutions, a − b and a + b; “less than” means between them, and “greater than” means outside them.
- An absolute value can never be negative, so |…| = −2 has no solution.
In this article
Absolute value is distance
The absolute value \(\displaystyle |x|\) is the distance from x to 0, so \(\displaystyle |5| = 5\) and \(\displaystyle |-5| = 5\). More useful on the SAT: \(\displaystyle |x - a|\) is the distance from x to a. Read \(\displaystyle |x - 2| = 3\) as “x is 3 units away from 2.” Two numbers fit: −1 and 5.
Use the number line below. Move the center a and the distance b, and switch between =, <, and >.
Try it yourself · interactive
Distance from a center
|x − 2| = 3
Solution: x = -1 or x = 5
Solve equations by splitting into two cases
If \(\displaystyle |\text{expression}| = b\) with b positive, the expression equals b or −b. For \(\displaystyle |2x - 4| = 6\): either \(\displaystyle 2x - 4 = 6\), so \(\displaystyle x = 5\), or \(\displaystyle 2x - 4 = -6\), so \(\displaystyle x = -1\). Check both in the original equation.
Isolate the absolute value first. In \(\displaystyle 3|x + 1| - 2 = 10\), add 2 and divide by 3 to get \(\displaystyle |x + 1| = 4\) before splitting.
Inequalities: between or outside
Distance makes inequalities intuitive. \(\displaystyle |x - 2| < 3\) means x is within 3 of 2, so \(\displaystyle -1 < x < 5\). \(\displaystyle |x - 2| > 3\) means x is more than 3 away, so \(\displaystyle x < -1\) or \(\displaystyle x > 5\).
- “Less than”: one combined interval, between a − b and a + b.
- “Greater than”: two separate pieces, below a − b or above a + b.
- Use ≤ or ≥ to include the endpoints.
The V-shaped graph
The graph of \(\displaystyle y = |x - h| + k\) is a V with its vertex at (h, k), just as vertex form works for a parabola. The graph below shows \(\displaystyle y = |x - 2| - 3\). Its x-intercepts, −1 and 5, are the solutions of \(\displaystyle |x - 2| = 3\).
The study lab
y = |x − 2| − 3
A V-shaped graph with its vertex at (2, −3).
- y = |x − 2| − 3
Your turn
Quick practice
What is the sum of the solutions to |3x − 1| = 8?
Hint: Find both solutions first, including the one from 3x − 1 = −8.
Practice: try these before checking the answers
Translate each one into a distance statement first.
- 1. Solve \(\displaystyle |x - 4| = 7\).
- 2. Solve \(\displaystyle |x + 1| < 3\).
- 3. Solve \(\displaystyle |2x| \ge 10\).
- 4. What is the vertex of the graph of \(\displaystyle y = |x + 5| + 2\)?
Sources & further reading
One step from here
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