Algebra
Grade 4-6Linear Inequalities
A linear inequality describes a whole range of possible values rather than a single answer, and is solved using almost the same steps as an equation, with one important extra rule to watch for. This lesson covers showing inequalities on a number line, solving a linear inequality, solving an inequality that involves dividing by a negative number, and listing the integer values that satisfy an inequality.
What you need to know
- Solve an inequality like an equation, but keep the inequality sign.
- If you multiply or divide both sides by a negative number, flip the sign.
- Show the solution on a number line: an open circle for a strict inequality, a filled circle for "or equal to".
- Integer solutions are the whole numbers that satisfy the inequality.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Showing an inequality on a number line
Use an open circle for a strict inequality (< or >), since the boundary value itself is not included, and a closed circle for ≤ or ≥, since the boundary value is included. Draw an arrow or line showing every value that satisfies the inequality.
To show x > 2, draw an open circle at 2 and an arrow extending to the right, since every value greater than 2 satisfies the inequality but 2 itself does not.
Solving a linear inequality
Solve an inequality using exactly the same steps as solving an equation, keeping the inequality sign instead of an equals sign throughout.
To solve 2x + 3 > 11, subtract 3 from both sides to get 2x > 8, then divide both sides by 2 to get x > 4.
Dividing an inequality by a negative number
When both sides of an inequality are multiplied or divided by a negative number, the inequality sign must be reversed.
To solve −2x + 5 > 13, subtract 5 from both sides to get −2x > 8. Dividing both sides by −2 reverses the inequality, giving x < −4.
Listing integer values that satisfy an inequality
Check each integer in the range against the inequality, remembering that a strict inequality (< or >) excludes the boundary value, while ≤ or ≥ includes it.
For −2 < x ≤ 3, x cannot equal −2, since the inequality is strict there, but x can equal 3, since ≤ includes the boundary. The integer values are −1, 0, 1, 2, 3.
Worked examples
Three exam-style questions, fully solved
Write down the inequality shown on a number line with an open circle at 3 and an arrow extending to the right.
Easy- 1.Identify the boundary value and whether it is included: an open circle at 3 means 3 is not included
- 2.Identify the direction: the arrow points right, towards larger values
Answer: x > 3
Solve 2x + 3 > 11.
Medium- 1.Subtract 3 from both sides: 2x > 8
- 2.Divide both sides by 2: x > 4
Answer: x > 4
Solve −2x + 5 > 13.
Hard- 1.Subtract 5 from both sides: −2x > 8
- 2.Divide both sides by −2, reversing the inequality: x < −4
Answer: x < −4
Practice
6 questions, marked instantly
Type an answer and check it. The worked solution appears once you have had a go. No login needed, and your progress saves in this browser.
Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
Solve x + 5 > 12.
Solve 3x ≤ 15.
Solve 2x − 1 < 9.
Solve 4x + 3 ≥ x + 12.
Solve −2x > 6.
List all the integers that satisfy −2 ≤ x < 3.
Avoid these
The mistakes students make most often
Using a closed circle for a strict inequality (< or >) instead of an open circle, or the reverse for ≤ or ≥.
Forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number.
Including or excluding the boundary value incorrectly when listing integer solutions, especially when the inequality is strict at one end and not the other.
Drawing the arrow on a number line pointing in the wrong direction relative to the inequality symbol.
Solving an inequality exactly like an equation all the way through, without checking whether a step, such as dividing by a negative, requires the sign to be reversed.
FAQ
Questions students and parents ask
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