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Linear Inequalities

2.0 Algebra · 2.3 Linear Inequalities

Syllabus tag: Kenya CBC | Grade 7 Mathematics | Strand 2.0 Algebra | Sub-Strand 2.3 Linear Inequalities (8 lessons)

Lesson objectives

By the end of this topic, you will be able to:

  • Apply inequality symbols to inequality statements
  • Form and illustrate simple linear inequalities in one unknown on a number line
  • Form and illustrate compound inequality statements in one unknown on a number line
  • Appreciate the use of linear inequalities in real life

Linear Inequalities

An equation says two things are equal. An inequality says one is bigger or smaller than the other.

Most real limits are inequalities. A lift carries at most eight people. A pass mark is at least fifty.

a) The four symbols

The symbol < means less than, and > means greater than. Neither includes the number itself.

The symbol ≤ means less than or equal to, and ≥ means greater than or equal to. Both include the number itself.

So x < 5 does not allow x to be 5, but x ≤ 5 does.

Read the symbol from the wide end. The wide end faces the larger quantity.

b) On a number line

x is greater than 2 x is greater than 2 −3 −2 −1 0 1 2 3 4 5 6 7 8

An open circle means the number is not included. This is x > 2, so 2 itself is left out.

x is at most 5 x is at most 5 −3 −2 −1 0 1 2 3 4 5 6 7 8

A filled circle means the number is included. This is x ≤ 5, so 5 counts.

Decide the circle from the symbol before you draw it. Getting it wrong changes the answer.

c) Compound inequalities

A compound inequality traps the letter between two values.

−2 < x ≤ 3 −2 < x ≤ 3 −5 −4 −3 −2 −1 0 1 2 3 4 5 6

Read it in two halves. −2 < x ≤ 3 says x is greater than −2 and at most 3.

Each end is drawn separately. Here −2 is open because of the <, and 3 is filled because of the ≤.

The whole segment between them is shaded, because every value in between satisfies both conditions.

d) Solving an inequality

Solve it exactly as you would an equation. Do the same thing to both sides until the letter is alone.

Solving an inequality Solving an inequality 2x + 3 < 11 the inequality 2x < 8 subtract 3 from both sides x < 4 divide both sides by 2

The answer is not one number. It is every number below 4.

e) Writing the answer

Give the answer as an inequality, not as a single value.

Then show it on a number line if the question asks, with the correct circle at the end.

Check by testing a number from your answer. For x < 4, try x = 1. Then 2(1) + 3 = 5, which is indeed less than 11.

f) Where this is used

A matatu has a maximum number of passengers. A shop offers a discount above a minimum spend. A bridge has a weight limit. Each of these is an inequality.

Words to know

  • Inequality — a mathematical statement that one expression is greater than, less than, or not equal to another.
  • Compound inequality — an inequality combining two conditions on the same unknown, giving both a lower and upper boundary.
  • Open circle (on a number line) — a mark showing that a boundary value is not included in the solution.
  • Filled circle (on a number line) — a mark showing that a boundary value is included in the solution.

:::checkpoint Check yourself

  1. What is the difference between x < 7 and x ≤ 7?
  2. Show x ≥ −1 on a number line. Which circle do you use?
  3. Solve 3x − 2 < 13.
  4. Write in symbols: y is greater than 0 and at most 6. :::

Bridge to practice

Try the exercises below — on forming, solving and illustrating simple and compound inequalities.

Check yourselfPractise Linear Inequalities — 10 questions →Next in MathematicsFractions →