Equations and inequalities

Inequalities

Inequalities are solved almost like equations. One rule makes the difference, and the answer is a range of values instead of a single number.

The rules

• Adding or subtracting on both sides keeps the inequality sign.
• Multiplying or dividing by a negative number flips the sign: −2x < 6 gives x > −3.
• Quadratic inequalities: find the zeros, then check which intervals satisfy the inequality.
• Write the solution as an interval or set, for example x < 4 or ]−∞; 4[.

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 4x − 5 < 1. What is the largest integer x that works?

   • a6
   • b0
   • c1
   • d2
   Show solution

   Answer: 1

   Isolate x. Dividing by a positive number keeps the < direction.

   x < (1 + 5) / 4 = 1.5. Largest integer below that: 1.

2. 2. Solve 3x + 5 < 25. What is the largest integer x that works?

   • a20
   • b7
   • c5
   • d6
   Show solution

   Answer: 6

   Isolate x. Dividing by a positive number keeps the < direction.

   x < (25 − 5) / 3 = 20/3. Largest integer below that: 6.

3. 3. Solve x² + 3x − 4 ≤ 0. How many integers x satisfy it?

   • a6
   • b5
   • c7
   • d4
   Show solution

   Answer: 6

   Find the zeros; an upward parabola is below the x-axis between them.

   Zeros -4 and 1, so -4 ≤ x ≤ 1: 6 integers.

4. 4. Solve 4x − 1 < 5. What is the largest integer x that works?

   • a2
   • b6
   • c0
   • d1
   Show solution

   Answer: 1

   Isolate x. Dividing by a positive number keeps the < direction.

   x < (5 + 1) / 4 = 1.5. Largest integer below that: 1.

5. 5. Solve 4x − 1 < 30. What is the largest integer x that works?

   • a31
   • b6
   • c7
   • d8
   Show solution

   Answer: 7

   Isolate x. Dividing by a positive number keeps the < direction.

   x < (30 + 1) / 4 = 7.75. Largest integer below that: 7.

6. 6. Solve 4x − 4 < 22. What is the largest integer x that works?

   • a5
   • b7
   • c6
   • d26
   Show solution

   Answer: 6

   Isolate x. Dividing by a positive number keeps the < direction.

   x < (22 + 4) / 4 = 6.5. Largest integer below that: 6.

Common mistakes

• Not flipping the sign after dividing by a negative number.
• Giving the zeros of a quadratic inequality instead of the interval.
• Including a boundary value when the sign is strict.

Related topics

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