Equations and inequalities

Absolute value equations

The absolute value |x| is the distance of x from zero, so it is never negative. Equations with it split into two cases.

The rules

• |a| = a if a ≥ 0 and |a| = −a if a < 0.
• |x − c| = d with d > 0 has two solutions: x = c + d and x = c − d.
• |x − c| = d with d < 0 has no solution.
• For harder cases, solve each case separately and keep only solutions in that case's range.

Practice questions with solutions

Generated by the same engine as the Kollego course and solvable without a calculator. Pick an answer in your head, then open the solution.

1. 1. Solve |x| < 6. What is the width of the solution interval?

   • a12
   • b6
   • c11
   • d13
   Show solution

   Answer: 12

   |x − a| < b means a − b < x < a + b, an interval of width 2b.

   Solution: -6 < x < 6. Width = 6 − (-6) = 12.

2. 2. Solve |x + 9| = 3. Give the smaller solution.

   • a-11
   • b-12
   • c-13
   • d-6
   Show solution

   Answer: -12

   |x − a| = b means x − a = b or x − a = −b, giving x = a + b or x = a − b.

   x + 9 = + 3 or x + 9 = − 3 → x = -6 or x = -12.

3. 3. Solve |x + 5| = 6. What is the sum of the two solutions?

   • a-10
   • b-11
   • c12
   • d-9
   Show solution

   Answer: -10

   |x − a| = b has two solutions, a + b and a − b, symmetric about a; their sum is 2a.

   x = -5 + 6 or x = -5 − 6, i.e. x = 1 or x = -11. Sum = -10.

4. 4. Solve |x − 6| = 5. What is the sum of the two solutions?

   • a11
   • b12
   • c13
   • d10
   Show solution

   Answer: 12

   |x − a| = b has two solutions, a + b and a − b, symmetric about a; their sum is 2a.

   x = 6 + 5 or x = 6 − 5, i.e. x = 11 or x = 1. Sum = 12.

5. 5. Solve |x − 3| < 7. What is the (exclusive) upper bound of the solution interval?

   • a-4
   • b10
   • c7
   • d3
   Show solution

   Answer: 10

   |x − a| < b means a − b < x < a + b.

   -4 < x < 10, so the upper bound is 10.

6. 6. Solve |x − 9| < 6. What is the (exclusive) upper bound of the solution interval?

   • a9
   • b15
   • c6
   • d3
   Show solution

   Answer: 15

   |x − a| < b means a − b < x < a + b.

   3 < x < 15, so the upper bound is 15.

Common mistakes

• Giving only one of the two solutions.
• Accepting a solution from a case where it does not lie in the range.
• Forgetting that an absolute value cannot equal a negative number.

Related topics

All maths topics · T-Kurs · M-Kurs · W-Kurs