Chapter 09

Inequalities and one-step equations

A one-step equation has exactly one thing done to the letter, and it is solved by undoing that one thing on both sides. Addition is undone by subtraction, multiplication by division.

An inequality is solved the same way, with one exception worth learning early: multiplying or dividing by a negative number reverses which side is larger, so the sign has to turn around with it.

Inequalities and one-step equations

An inequality says one side is larger or smaller than the other rather than equal to it. A one-step equation is solved by undoing the single operation applied to the letter.

  • < means less than and > means greater than: 3 < 7.
  • ≤ means less than or equal to, and ≥ means greater than or equal to.
  • x + 8 = 15 is solved by subtracting 8 from both sides, so x = 7.
  • 4x = 20 is solved by dividing both sides by 4, so x = 5.
  • The same undoing works on an inequality: x + 3 > 10 gives x > 7.
  • Multiplying or dividing an inequality by a negative number flips the sign: −2x < 6 gives x > −3.

Solving x − 5 ≥ 2

Add 5 to both sides.

x ≥ 2 + 5

x ≥ 7

Examples

Each one is set out the way a solution is written on paper: the question, the steps in order, then the answer.

Example 1. One-step equations

Solve x + 8 = 15, then x − 5 = 9, then 4x = 20.

Solution

  1. x + 8 = 15: subtract 8 from both sides, so x = 7.
  2. x − 5 = 9: add 5 to both sides, so x = 14.
  3. 4x = 20: divide both sides by 4, so x = 5.

x = 7, x = 14, x = 5

Example 2. A two-step equation

Solve 2x + 5 = 17.

Solution

  1. Subtract 5 from both sides: 2x = 12.
  2. Divide both sides by 2: x = 6.
  3. Check it: 2(6) + 5 = 17. It works.

x = 6

Example 3. Another two-step equation

Solve 3x − 4 = 11.

Solution

  1. Add 4 to both sides: 3x = 15.
  2. Divide both sides by 3: x = 5.
  3. Check it: 3(5) − 4 = 11. It works.

x = 5

Example 4. An inequality that flips

Solve −3x ≤ 12.

Solution

  1. Divide both sides by −3.
  2. Dividing by a negative number reverses which side is larger, so ≤ becomes ≥.
  3. x ≥ −4.

x ≥ −4

Where this leads

The step-by-step solver
It solves one-step and two-step equations and prints every line of the working.