Modulus Function: Absolute Value Equations and Inequalities

What Is the Modulus of a Number?

The modulus (or absolute value) of a number is its size without its sign. It is written with vertical bars: |5| = 5, |−5| = 5 and |0| = 0. Geometrically, |x| is the distance of x from 0 on the number line, and |x − a| is the distance between x and a. This tutorial covers the modulus function, its graph, how to solve modulus equations and inequalities, and the common mistakes that cost marks in WAEC Further Mathematics, A-level-style and first-year university questions.

Definition of the Modulus Function

The modulus function is defined in two pieces:

|x| = x when x ≥ 0;
|x| = −x when x < 0.

The second line often confuses students: when x is negative, −x is positive. For example, |−7| = −(−7) = 7.

Useful properties:

|x| ≥ 0 for every x, and |x| = 0 only when x = 0.
|ab| = |a| × |b| and |a/b| = |a| / |b| (b ≠ 0).
|x|² = x², and √(x²) = |x|, not x. For example √((−3)²) = √9 = 3 = |−3|.
The triangle inequality: |a + b| ≤ |a| + |b|. Note that |a + b| is generally not equal to |a| + |b|: |3 + (−5)| = 2, but |3| + |−5| = 8.

Graphs of the Modulus Function

The graph of y = |x| is a V shape with its vertex at the origin: the line y = x for x ≥ 0, and the line y = −x for x < 0. It has gradient 1 on the right, −1 on the left, and a sharp corner at 0.

y = |ax + b|. Sketch y = ax + b, then reflect any part below the x-axis in the axis. For y = |2x − 4|, the line crosses the axis at x = 2, so the graph is a V with vertex at (2, 0) and a y-intercept of |−4| = 4.

y = |f(x)|. The same rule works for any function: parts of the graph below the x-axis are reflected upwards. For y = |x² − 4|, the part of the parabola between x = −2 and x = 2 (which dips to −4) is flipped to make a hump reaching 4.

y = f(|x|). This is different: take the part of y = f(x) for x ≥ 0 and reflect it in the y-axis, so the graph is symmetric. For y = |x|² − 4|x|, the right half is the curve x² − 4x, and the left half mirrors it.

Solving Modulus Equations

The basic rule: |X| = k (with k ≥ 0) means X = k or X = −k. If k < 0 there is no solution, because a modulus can't be negative.

Example 1. Solve |2x − 3| = 7.

2x − 3 = 7 gives x = 5. 2x − 3 = −7 gives x = −2. Answer: x = 5 or x = −2. Check: |10 − 3| = 7 ✓ and |−4 − 3| = 7 ✓.

Example 2. Solve |x + 4| = −2.

No solution: a modulus can never equal a negative number.

Example 3 (modulus on both sides). Solve |x − 1| = |2x + 3|.

|A| = |B| means A = B or A = −B. Alternatively, square both sides, since both are non-negative.

x − 1 = 2x + 3 gives x = −4.
x − 1 = −(2x + 3) gives 3x = −2, so x = −2/3.

Answer: x = −4 or x = −2/3. Check x = −4: |−5| = 5 and |−5| = 5 ✓.

Equations Where Checking Is Essential

When the right-hand side contains x, some answers may not work, so always check.

Example 4. Solve |x − 2| = 3x + 6.

Case 1: x − 2 = 3x + 6 gives −2x = 8, so x = −4. Check: the left side is |−6| = 6, the right side is 3(−4) + 6 = −6. They are not equal, so reject x = −4.
Case 2: x − 2 = −(3x + 6) gives 4x = −4, so x = −1. Check: |−3| = 3 and 3(−1) + 6 = 3 ✓.

Answer: x = −1 only. The rejected value made the right side negative, which a modulus can never equal. A quick rule: any solution must make the right-hand side ≥ 0.

Example 5. Solve |x² − 5| = 4.

x² − 5 = 4 gives x² = 9, so x = ±3. x² − 5 = −4 gives x² = 1, so x = ±1. Answer: x = −3, −1, 1 or 3. On the graph of y = |x² − 5|, the horizontal line y = 4 crosses it four times.

Solving Modulus Inequalities

Think of the modulus as distance.

|X| < k means X is within k of 0: −k < X < k.
|X| > k means X is more than k from 0: X < −k or X > k.

Example 6. Solve |x − 3| < 5.

−5 < x − 3 < 5. Add 3 throughout: −2 < x < 8. In words: x is less than 5 units from 3.

Example 7. Solve |2x + 1| ≥ 9.

2x + 1 ≤ −9 or 2x + 1 ≥ 9. So 2x ≤ −10 or 2x ≥ 8. Answer: x ≤ −5 or x ≥ 4.

Example 8. Solve |x − 1| > |x + 3|.

Both sides are non-negative, so squaring keeps the inequality the same:

(x − 1)² > (x + 3)², so x² − 2x + 1 > x² + 6x + 9, so −8x > 8, so x < −1.

Distance interpretation: x is further from 1 than from −3, which is true for every point to the left of the midpoint −1.

Example 9. Solve |x − 2| < 2x − 1.

Split into cases using the definition.

If x ≥ 2: |x − 2| = x − 2, so x − 2 < 2x − 1, giving x > −1. Combined with x ≥ 2, this gives x ≥ 2.
If x < 2: |x − 2| = 2 − x, so 2 − x < 2x − 1, giving 3 < 3x, so x > 1. Combined with x < 2, this gives 1 < x < 2.

Putting the cases together: x > 1. Check x = 1.5: |−0.5| = 0.5 < 2 ✓; check x = 0: 2 < −1 is false ✓.

Errors and Tolerances

Modulus inequalities describe tolerances. A machine part should be 50 mm long with a tolerance of 0.2 mm. The acceptable lengths L satisfy |L − 50| ≤ 0.2, that is 49.8 ≤ L ≤ 50.2. In statistics, |x − μ| < 2σ describes values within two standard deviations of the mean. In numerical methods, an iteration stops when |xₙ₊₁ − xₙ| is less than a chosen tolerance.

Summary

|x| is the distance of x from 0; |x − a| is the distance between x and a.
|X| = k (k ≥ 0) gives X = k or X = −k. |A| = |B| gives A = B or A = −B.
When the other side contains x, check every answer, and reject any that make it negative.
|X| < k gives −k < X < k; |X| > k gives X < −k or X > k.
For |A| compared with |B|, squaring both sides is safe because both are non-negative.
Graph y = |f(x)| by reflecting the negative parts of y = f(x) in the x-axis.