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 17
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 17
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
Problem 3 of 17
|x − 1| = 6. What is the larger solution?
Problem 4 of 17
|x + 3| = 2. What is the smaller solution?
Problem 5 of 17
|x + 2| − 3 = 5. What is the larger solution?
Problem 6 of 17
|2x + 1| = 7. What is the smaller solution?
Problem 7 of 17
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.
Problem 8 of 17
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
Problem 9 of 17
What is the solution Dev missed?
Problem 10 of 17
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
Problem 11 of 17
With the absolute value isolated first, what is the smaller solution?
Problem 12 of 17
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
Problem 13 of 17
What should line 4 say?
Pick the Move
Problem 14 of 17
5|x − 2| + 4 = 19. What is the first move?
Problem 15 of 17
|x + 6| = −2. What is the first move?
Same Solution?
Without solving all the way, decide which equations share both answers.
Problem 16 of 17
Which of these have the same two solutions as |x − 3| = 5? Select every one.
Problem 17 of 17
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.