Part 24 of 36 · InequalitiesSee the whole course
Absolute value equations and graphs
The absolute value |x| is how far x is from 0, so it is never negative: |5| = 5 and |−5| = 5. To solve |something| = k, with k zero or more, solve two equations: something = k and something = −k. If k is negative there is no solution. The graph of y = |x| is a V with its point at the origin.
What absolute value means
Picture a number line. The absolute value of a number is its distance from 0, counted in steps and never in a direction. 5 is five steps from 0, and so is −5, so |5| = |−5| = 5. And |0| = 0.
It is not just “drop the minus sign”. Bars work on the whole result inside: |3 − 8| = |−5| = 5. Do the sum first, then take the distance.
The graph is a V
For x zero or more, |x| is just x, so the graph follows the line y = x. For x less than zero, |x| flips the sign, so it follows y = −x. Together they make a V with its point (the vertex) at the origin, and both arms have slope 1 and −1.
Change x to x − h and the V slides sideways so that its point is at x = h. This is the same move as in the horizontal shift of a function.
Solving means cutting the V with a line
To solve |x − h| = k, draw the horizontal line y = k across the V. The x-values where they meet are the solutions. A line cuts a V in two places, and the two places are the same distance from h, one on each side. So the solutions are x = h + k and x = h − k. Try it:
- Case 1: x − 2 = 3, so x = 5
- Case 2: x − 2 = −3, so x = −1
Example 1: |x| = 5
Two numbers are 5 steps from 0.
Case 1: x = 5
Case 2: x = −5
The solutions are x = 5 or x = −5.
Example 2: |x − 3| = 4
The number x − 3 is 4 steps from 0, so it is 4 or −4.
Case 1: x − 3 = 4, so x = 7
Case 2: x − 3 = −4, so x = −1
Check: |7 − 3| = 4 ✓ and |−1 − 3| = |−4| = 4 ✓
The solutions are x = 7 or x = −1: the two numbers that are 4 away from 3.
Example 3: with something inside, |2x + 1| = 7
Case 1: 2x + 1 = 7, so 2x = 6 and x = 3
Case 2: 2x + 1 = −7, so 2x = −8 and x = −4
Check: |2(3) + 1| = 7 ✓ and |2(−4) + 1| = |−7| = 7 ✓
The solutions are x = 3 or x = −4.
Example 4: get the absolute value alone first, 2|x − 1| + 3 = 11
You can only split into two cases once the bars stand alone on one side.
Subtract 3: 2|x − 1| = 8
Divide by 2: |x − 1| = 4
Case 1: x − 1 = 4, so x = 5
Case 2: x − 1 = −4, so x = −3
Check x = −3: 2|−4| + 3 = 8 + 3 = 11 ✓
The solutions are x = 5 or x = −3.
One solution, or none
- |x − 2| = 0 has one solution, x = 2. The line y = 0 only touches the point of the V.
- |x| = −3 has no solution. A distance is never negative, and the line y = −3 is below the V and never meets it. Try k = −1 in the graph above.
A peek at inequalities
The same picture solves inequalities. |x| < 3 means “less than 3 steps from 0”, which is −3 < x < 3 (the part of the V below the line y = 3). |x| > 3 means “more than 3 steps away”, which is x < −3 or x > 3 (the part above it). If inequalities are new, start with how to solve a linear inequality.
Watch it explained in under a minute
Read the video transcript
Absolute value means distance from zero. 5 is 5 steps away. So is −5.
Graph y = |x|. It makes a V, with its point at 0.
Now solve: |x| = 5. Draw the line y = 5. It crosses the V twice. So x = 5, or x = −5.
Slide the V 3 to the right. Now it is |x − 3|. Set it equal to 4. x = 7, or x = −1.
Want practice questions and more examples? Full guide: pipmath.app/blog
The five mistakes to avoid
- Giving only one solution. Most absolute value equations have two.
- Splitting the equation before the absolute value is alone. Subtract and divide first.
- Getting the second case wrong. It is x − 3 = −4, not x + 3 = 4.
- Not noticing that |something| = a negative number has no solution.
- Thinking |−5| is −5. Absolute value is never negative.
Try it yourself
Solve each equation. Tap a question to see the answer.
1. |x| = 8
The numbers 8 steps from 0: x = 8 or x = −8.
2. |x + 2| = 5
Case 1: x + 2 = 5, so x = 3. Case 2: x + 2 = −5, so x = −7. x = 3 or x = −7. Check: |3 + 2| = 5 ✓ and |−7 + 2| = |−5| = 5 ✓.
3. |2x − 4| = 6
Case 1: 2x − 4 = 6, so 2x = 10 and x = 5. Case 2: 2x − 4 = −6, so 2x = −2 and x = −1. x = 5 or x = −1. Check: |10 − 4| = 6 ✓ and |−2 − 4| = 6 ✓.
Where this fits in
The V is one of the most useful shapes for seeing transformations: shift it, flip it and stretch it. That is the next part of the course, starting with function transformations.