The absolute value of a number is its distance from zero, so it is never negative. |x| = 5 means x is 5 units from zero, and that happens twice: at 5 and at −5.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Which numbers are exactly 4 away from 3 on the number line? Select every one.
3
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Two cases
|x − 2| = 6 says the distance between x and 2 is 6. Either x − 2 = 6, so x = 8, or x − 2 = −6, so x = −4. One equation, two cases, two answers, and both check.
Isolate first
2|x + 1| − 3 = 7 must be tidied before the split: add 3, 2|x + 1| = 10; divide by 2, |x + 1| = 5. Only now do the two cases appear: x + 1 = 5 or x + 1 = −5. Splitting earlier gives the wrong cases.
Exactly one answer
|x − 3| = 0 has one solution, x = 3, because only zero is at distance zero from zero. The two cases collapse into one.
When there is no answer
|x + 4| = −3 has no solution, whatever x is. A distance cannot be negative. Recognize it before writing cases that lead nowhere.
Check both
Substitute each answer into the original. |8 − 2| = 6 and |−4 − 2| = |−6| = 6. Both hold. A case that fails the check was a slip in the arithmetic, and it happens most often in the negative case.
Step 2: Try It Yourself
Tap and try it out.
Solve 2|x + 1| − 3 = 7
- 12|x + 1| − 3 = 7the equation as given
Step 0 of 5
5
Step 3: In Real Life
A factory tolerance
A bolt must be 10 millimeters, within 0.2: |d − 10| = 0.2 marks the limits. Two answers, 9.8 and 10.2, because a distance can be on either side.
Step 4: Watch an Example
One step at a time.
Watch Ana Solve |2x − 3| = 7
The absolute value is already alone, and there is a 2x inside it.
- Step 1
The bars are already alone on the left, so there is nothing to isolate. The equation says 2x − 3 is 7 from zero.
Watch Ravi Solve 3|x + 1| − 2 = 10
The absolute value is not alone yet. The last three steps are yours before they appear.
- Step 1
The bars are multiplied by 3 and have 2 taken away, so the absolute value is not alone. Adding 2 to both sides gives 3|x + 1| = 12.
Two Roads to |x − 5| = 3
One equation, solved by a picture and by cases.
- Step 1
Road one is the number line. |x − 5| = 3 says the distance between x and 5 is 3. Three steps right of 5 lands on 8; three steps left lands on 2.
Step 5: Your Turn
Practice makes it stick.
The Filling Line
Problem 1 of 2
A machine drops 400 g of dough into each tin and may be off by up to 6 g either way. The equation |m − 400| = 6 gives the two limits. What is the heavier limit, in grams?
g
The Thermostat
Problem 2 of 2
A thermostat is set to 68 °F. The heating switches on or off when the room is 3 degrees from the setting. The equation |t − 68| = 3 gives both switching temperatures. What is the lower one, in °F?
°F
Split and Solve
1 of 5
|x − 1| = 6. What is the larger solution?
2 of 5
|x + 3| = 2. What is the smaller solution?
3 of 5
|x + 2| − 3 = 5. What is the larger solution?
4 of 5
|2x + 1| = 7. What is the smaller solution?
5 of 5
3|x − 4| + 2 = 2. How many solutions?
Whose Step?
Each page has one line that does not follow from the one above. Find it, then fix it.
1 of 6
Dev solved |x − 3| = 5 and checked his answer. Which line is the slip?
Dev solves |x − 3| = 5
- 1|x − 3| = 5
- 2x − 3 = 5
- 3x = 8+ 3
- Answer|8 − 3| = 5, so x = 8
2 of 6
What is the solution Dev missed?
3 of 6
Maya solved 3|x − 2| + 1 = 13. Which line is the slip?
Maya solves 3|x − 2| + 1 = 13
- 13|x − 2| + 1 = 13
- 23(x − 2) + 1 = 13 or 3(x − 2) + 1 = −13
- 33x − 5 = 13 or 3x − 5 = −13
- 43x = 18 or 3x = −8+ 5
- Answerx = 6 or x = −8/3÷ 3
4 of 6
With the absolute value isolated first, what is the smaller solution?
5 of 6
Noor solved 2|x| + 8 = −2. Which line is the slip?
Noor solves 2|x| + 8 = −2
- 12|x| + 8 = −2
- 22|x| = −10− 8
- 3|x| = −5÷ 2
- Answerx = 5 or x = −5
6 of 6
What should line 4 say?
Pick the Move
1 of 2
5|x − 2| + 4 = 19. What is the first move?
2 of 2
|x + 6| = −2. What is the first move?
Same Solution?
Without solving all the way, decide which equations share both answers.
1 of 2
Which of these have the same two solutions as |x − 3| = 5? Select every one.
2 of 2
Which of these have the same two solutions as |x| = 4? Select every one.
Step 6: Quick Check
Show what you know.
Question 1 of 1
|x − 4| = 3. What is the larger solution?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
An absolute value equation |x − a| = b has the solutions 2 and 10. What is a?
Stretch 2 of 2
For which value of k does |x − 4| = k have exactly one solution?
What You Learned
- Absolute value is distance from zero, and is never negative.
- Isolate the absolute value, then split into two cases.
- Equal to zero gives one solution; equal to a negative gives none.
- Check both answers in the original.
For the grown-up
The slip this lesson targets is dropping the negative case: |x − 3| = 5 read as x − 3 = 5 alone finds x = 8, and a check in the original passes, because 8 is a solution. A check confirms an answer; it cannot find a missing one. Ask how many numbers are 5 away from 3.