Absolute value

The absolute value of a number is its distance from zero on the number line. Distance is never negative, so absolute value is always positive or zero. It is written with two vertical bars: |−7| = 7 and |7| = 7, because both are 7 steps from zero. In short, absolute value keeps the size of a number and drops its sign.

What is absolute value?

−4 and 4 are opposites: the same distance from zero on different sides. They have the same absolute value, 4. The US Common Core introduces this in grade 6, using real examples: a bank balance of −30 dollars is a debt whose size is |−30| = 30 dollars. Absolute value is also called the modulus, the usual word in UK A-level Maths.

The definition

In algebra, absolute value is defined in two parts:

The second line looks odd: how can −x be positive? Because x itself is negative there. If x = −4, then −x = −(−4) = 4.

Absolute value and order of operations

Treat the bars like brackets: work out what is inside first, then take the absolute value. That makes a difference:

Distance between two numbers

The distance between two numbers a and b on a number line is |a − b|. The distance between −3 and 5 is |−3 − 5| = |−8| = 8, and it comes out the same the other way round: |5 − (−3)| = 8. Grade 6 Common Core uses this to find distances between points with the same x or y coordinate.

Solving absolute value equations

If |x| = 5, then x is 5 steps from zero, so x = 5 or x = −5. Every absolute value equation works the same way:

1. Get the absolute value on its own on one side.
2. Check the other side. If it is negative, there is no solution. If it is zero, there is one.
3. Split into two equations: the inside equals the positive value, or the inside equals the negative value.
4. Solve both and check them in the original.

Example 1: |x − 3| = 5

Split: x − 3 = 5 or x − 3 = −5. So x = 8 or x = −2. Check: |8 − 3| = 5 and |−2 − 3| = 5.

Example 2: |2x + 1| = 7

Split: 2x + 1 = 7 gives x = 3, and 2x + 1 = −7 gives x = −4. So x = 3 or x = −4.

Example 3: isolate it first

Solve 2|x + 4| − 3 = 9. Add 3 to get 2|x + 4| = 12, then divide by 2: |x + 4| = 6. Split: x + 4 = 6 or x + 4 = −6, so x = 2 or x = −10.

Example 4: no solution

|x + 1| = −4 has no solution. An absolute value can never equal a negative number, so stop there.

Absolute value inequalities

Think distance again. |x| < 3 means x is less than 3 steps from zero, so it is between −3 and 3. |x| > 3 means more than 3 steps away, so it is beyond 3 in either direction.

For |x − 2| ≤ 5, write −5 ≤ x − 2 ≤ 5, then add 2 to all three parts. For |x − 2| ≥ 4, solve x − 2 ≤ −4 and x − 2 ≥ 4 separately. Either way, you finish with ordinary linear inequalities. Practise one of those now, then press Mark to check it in the course.

The absolute value graph y = |x|

Here x is any number and y is its absolute value. The graph is a V shape with its point, the vertex, at the origin. Each side is a straight line: y = x on the right and y = −x on the left. It never dips below the x-axis, because absolute values are never negative.

Changing the equation moves the V. y = |x − 2| − 1 has its vertex at (2, −1): the −2 inside shifts it 2 right, and the −1 outside shifts it 1 down. The Graphing Functions course covers shifts like these for every kind of graph.

Its two x-intercepts, at 1 and 3, are exactly the solutions of |x − 2| = 1: the graph is zero where |x − 2| equals 1.

Common mistakes

• Dropping signs inside the bars too early. |3 − 8| is |−5| = 5, not 3 + 8 = 11.
• Splitting before isolating. In 2|x + 4| − 3 = 9, get |x + 4| alone first.
• Giving only one answer. Most absolute value equations have two solutions.
• Solving |x| = −4. There is no solution, so don't produce one.
• Mixing up "and" and "or". Less than gives one interval between two values. Greater than gives two separate pieces.

To practise these skills among other topics in a short timed paper, try today's Pre-Algebra & Algebra 1 or Geometry, Algebra 2 & SAT Math paper. For negative numbers more generally, see integers.

Read next

Maths · AlgebraSet notation in A-level maths: symbols and inequalitiesThe set notation symbols you need for A-level maths, how to write the solution of an inequality in set notation, and how sets are used in probability.