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Triangle Inequality with Absolute Value

Triangle Inequality with Absolute Value

An alternate version of the triangle inequality.

 

              Triangle Inequality:     |a + b| ≤ |a| + |b|

Alternate Triangle Inequality:     |a – b| ≥ |a| – |b|

 

 

See also

Absolute value, absolute value rules

Key Formula

∣a+b∣≤∣a∣+∣b∣|a + b| \leq |a| + |b|
Where:
  • aa = Any real number
  • bb = Any real number

Worked Example

Problem: Verify the triangle inequality |a + b| ≤ |a| + |b| for a = −7 and b = 3.
Step 1: Compute a + b.
a+b=−7+3=−4a + b = -7 + 3 = -4
Step 2: Take the absolute value of the sum.
∣a+b∣=∣−4∣=4|a + b| = |-4| = 4
Step 3: Compute |a| + |b| separately.
∣a∣+∣b∣=∣−7∣+∣3∣=7+3=10|a| + |b| = |-7| + |3| = 7 + 3 = 10
Step 4: Compare the two sides. The left side (4) is indeed less than or equal to the right side (10), so the inequality holds.
4≤10✓4 \leq 10 \quad \checkmark
Answer: |a + b| = 4 and |a| + |b| = 10, so 4 ≤ 10 confirms the triangle inequality.

Another Example

Problem: Verify the alternate triangle inequality ||a| − |b|| ≤ |a − b| for a = −7 and b = 3.
Step 1: Compute a − b.
a−b=−7−3=−10a - b = -7 - 3 = -10
Step 2: Take the absolute value of the difference.
∣a−b∣=∣−10∣=10|a - b| = |-10| = 10
Step 3: Compute ||a| − |b||.
∣∣a∣−∣b∣∣=∣7−3∣=4\big||a| - |b|\big| = |7 - 3| = 4
Step 4: Compare. The lower bound (4) is indeed less than or equal to |a − b| (10).
4≤10✓4 \leq 10 \quad \checkmark
Answer: ||a| − |b|| = 4 and |a − b| = 10, so 4 ≤ 10 confirms the alternate triangle inequality.

Frequently Asked Questions

Why is it called the 'triangle' inequality if it's about absolute values?
The name comes from geometry: in any triangle, the length of one side is always less than or equal to the sum of the other two sides. Absolute values measure distance on the number line, so |a + b| ≤ |a| + |b| is the one-dimensional version of the same geometric principle. Going from the number line to higher dimensions, the same idea applies to vectors and gives the familiar triangle side-length rule.
When does equality hold in |a + b| = |a| + |b|?
Equality holds exactly when a and b have the same sign (or at least one of them is zero). In that case, no cancellation occurs in the sum, so the magnitude of a + b equals the total of the individual magnitudes. For example, |3 + 5| = |3| + |5| = 8, but |−3 + 5| = 2 < 8.

Triangle Inequality: |a + b| ≤ |a| + |b| vs. Alternate (Reverse) Triangle Inequality: ||a| − |b|| ≤ |a − b|

The standard form gives an upper bound on the absolute value of a sum. The alternate form gives a lower bound on the absolute value of a difference. Together they sandwich distances: you know |a − b| is at least ||a| − |b|| and |a + b| is at most |a| + |b|. The alternate form is actually derived from the standard form by a substitution argument.

Why It Matters

The triangle inequality with absolute value is one of the most frequently used inequalities in algebra, calculus, and analysis. It lets you place bounds on expressions without knowing exact values — for instance, bounding the error in an approximation. In proofs involving limits, convergence of sequences, and epsilon-delta arguments, this inequality is an essential tool for controlling the size of sums and differences.

Common Mistakes

Mistake: Writing |a + b| ≥ |a| + |b| (flipping the inequality direction).
Correction: The absolute value of a sum can never exceed the sum of the absolute values; it can only be less than or equal. Remember that partial cancellation can shrink the left side, never inflate it.
Mistake: Confusing the alternate form: writing |a − b| ≤ |a| − |b| instead of |a − b| ≥ ||a| − |b||.
Correction: The alternate form provides a lower bound, not an upper bound. Also, the right side must itself be wrapped in absolute value bars (||a| − |b||), because |a| − |b| could be negative.

Related Terms