➕ Algebra 1 · Inequalities
Absolute Value Equations: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
How far away is your friend?
A friend texts, “I live exactly 4 blocks from the library, on the same straight street.” That does not tell you which side of the library the friend lives on. Put the library at block 10. The friend lives at block 6 or at block 14. With x as the friend’s block number, |x − 10| = 4.
How it connects: A distance tells you how far, not which way, so a distance equation has two answers: |x − 10| = 4 gives x = 6 or x = 14. The limit: this works because everything is on one straight line, like a number line. On a flat map, a whole circle of spots is the same straight-line distance from the library, not just two.
The cone on the track
A coach puts a cone at the 50-meter mark of a straight 100-meter sprint lane and tells a runner to stop exactly 20 meters from the cone. The runner could stop at the 30-meter mark or at the 70-meter mark. With x as the runner’s position in meters from the start line, |x − 50| = 20, so x = 30 or x = 70.
How it connects: The cone’s position is the center, 20 meters is the distance, and the two stopping marks are the two solutions. The limit: it is only two spots because the runner stays in one straight lane. On an open field, every point on a circle around the cone is also 20 meters away.
Give or take
People say, “The movie is two hours long, give or take ten minutes.” “Give or take” means “plus or minus a small amount.” The movie could be 10 minutes shorter or 10 minutes longer than 120 minutes, so 110 or 130 minutes. With t as the length in minutes, |t − 120| = 10.
How it connects: “Give or take” is the tolerance: the target is 120 and the allowed difference is 10 either way. The equation gives the two edge values, 110 and 130. The limit: in everyday speech “give or take” is vague, and the amount may not be exact. A tolerance in a math or factory problem is an exact number, and all the allowed values together are described by an inequality, |t − 120| ≤ 10, which means the same as 110 ≤ t ≤ 130, not just by the two edges.
Measure twice, cut once
This carpenter’s proverb says to check your measurements carefully before you make a cut, because a cut cannot be undone.
How it connects: Checking both answers in the original equation is your second measurement. It catches arithmetic slips and the mistake of splitting before the bars were alone. The limit: check in the original equation, not in a line you rewrote. If your rewrite had a mistake, the check would agree with it and miss the problem.
Distance from sea level
Heights are measured from sea level, which is 0. A seagull flying 30 meters above sea level is at height 30. A fish swimming 30 meters below sea level is at height −30. Both are 30 meters from sea level: |30| = 30 and |−30| = 30. If all you know is that something is 30 meters from sea level, and h is its height in meters, then |h| = 30, so h = 30 or h = −30.
How it connects: Absolute value is the distance from 0, and it forgets which side of 0 you are on. That is why working backward from “30 meters away” leaves two possibilities. The limit: a height of −30 is a perfectly good number. It is only the distance, 30, that can never be negative.
The club for zero or more
A distance of −3 meters tried to enter the Absolute Value Club. The doorkeeper said, “Distances in here are zero or more. You are welcome back as 3.”
How it connects: A distance is never negative, so |x| = −3 has no solution. The limit: the joke is about distances. The number −3 itself is allowed on the number line, and its distance from 0 is |−3| = 3.
Memory help
BARS
Bars alone: get the absolute value by itself. Ask about the number on the other side: positive, zero or negative? Rewrite as two equations, inside = c and inside = −c, when c is positive (if c = 0 there is one equation, and if c is negative you stop: no solution). Solve both and check both in the original equation.
The count rhyme
“Positive, two. Zero, one. Negative, none: then the count is done.” Use it after the bars are alone, and only then.
Center and two arrows
For |x − c| = d with d > 0, mark c on a number line, draw an arrow of length d to the left and another to the right, and put a dot at each tip. The dots are the solutions. For |x + 1| = 4 the center is −1, so the dots are at 3 and −5.
Bars alone before you split
In 2|x + 7| + 1 = 9 the rule “inside equals plus or minus the right side” is not ready yet. First isolate: 2|x + 7| = 8, then |x + 7| = 4. Splitting into x + 7 = 9 or x + 7 = −9 gives wrong answers.
Negative on the right: stop
If the bars equal a negative number after you isolate them, there is no solution. Do not solve the equation anyway. In |x − 6| = −5, writing x − 6 = −5 gives x = 1, but |1 − 6| = 5, not −5.
How far, not which way
Absolute value answers “how far from 0?” and never “which way?” That is why working backward from a distance gives two ways to go (left or right), except when the distance is 0.