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Maths · Equations and inequalities

Solving linear inequalities

Solve linear inequalities, show solutions on a number line, list integer solutions, and at Higher tier show regions on a graph.

  • 6 key terms
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Warm-up

Answer each one, then check.

  1. 1

    Solve \(2x + 3 = 11\).

    Show answerHide answer

    \(x = 4\)

  2. 2

    Solve \(5x = -20\).

    Show answerHide answer

    \(x = -4\)

  3. 3

    Which is bigger, \(-3\) or \(-5\)?

    Show answerHide answer

    \(-3\)

  4. 4

    What is the symbol for "less than or equal to"?

    Show answerHide answer

    \(\le\)

  5. 5

    Solve \(\dfrac{x}{3} = 6\).

    Show answerHide answer

    \(x = 18\)

Learning Objectives

  1. 1Use the inequality symbols \(<\), \(>\), \(\le\) and \(\ge\).
  2. 2Show inequalities on a number line.
  3. 3Solve linear inequalities, including reversing the sign.
  4. 4Write down integer solutions and solve double inequalities.

Inequality Symbols

The open end of the symbol points at the bigger number.

  • \(x > 3\)

    Meaning: x is greater than 3. Circle on a number line: Open circle at 3

  • \(x \ge 3\)

    Meaning: x is greater than or equal to 3. Circle on a number line: Filled circle at 3

  • \(x < 3\)

    Meaning: x is less than 3. Circle on a number line: Open circle at 3

  • \(x \le 3\)

    Meaning: x is less than or equal to 3. Circle on a number line: Filled circle at 3

Solving an Inequality

Solve \(3x + 2 \le 14\).

Show the solutionHide the solution
  1. 1 Subtract 2 from both sides \(3x \le 12\)
  2. 2 Divide both sides by 3 \(x \le 4\)

Answer\(x \le 4\)

THE ONE NEW RULE

When you multiply or divide an inequality by a negative number, reverse the inequality sign.

Otherwise solve exactly as you would an equation.

A Negative Coefficient

Solve \(5 - 2x > 11\).

Show the solutionHide the solution
  1. 1 Subtract 5 from both sides \(-2x > 6\)
  2. 2 Divide by \(-2\) and reverse the sign \(x < -3\)
  3. 3 Check with a value, \(x = -4\) \(5 - 2(-4) = 13 > 11\)

Answer\(x < -3\)

A Double Inequality

Solve \(5 < 2x - 1 \le 9\).

Show the solutionHide the solution
  1. 1 Add 1 to all three parts \(6 < 2x \le 10\)
  2. 2 Divide all three parts by 2 \(3 < x \le 5\)

Answer\(3 < x \le 5\)

Integer Solutions

List the integers that satisfy \(-2 < x \le 3\).

Show the solutionHide the solution
  1. 1 \(-2\) itself is not included Start at \(-1\)
  2. 2 3 is included End at 3

Answer\(-1, 0, 1, 2, 3\)

Showing an Inequality on a Graph

Draw the boundary, then decide which side to shade.

  1. 1 Draw the line

    Replace the inequality sign with = and draw it
  2. 2 Solid or dashed

    Solid for \(\le\) and \(\ge\); dashed for \(<\) and \(>\)
  3. 3 Test a point

    For example \((0, 0)\), if it is not on the line
  4. 4 Shade the side that works

    Shade the wanted region, or the unwanted region, as the question asks

A Region from Three Inequalities

A region R satisfies \(y \ge 1\), \(y \le x\) and \(x + y \le 6\). Find the vertices of R.

Show the solutionHide the solution
  1. 1 \(y = 1\) meets \(y = x\) \((1, 1)\)
  2. 2 \(y = 1\) meets \(x + y = 6\) \((5, 1)\)
  3. 3 \(y = x\) meets \(x + y = 6\) \(x + x = 6\), so \((3, 3)\)

AnswerR is a triangle with vertices \((1, 1)\), \((5, 1)\) and \((3, 3)\).

A Quadratic Inequality

Solve \(x^2 - x - 6 < 0\).

Show the solutionHide the solution
  1. 1 Factorise \((x - 3)(x + 2) < 0\)
  2. 2 The roots are \(x = 3\) and \(x = -2\)
  3. 3 The graph is U-shaped, below the axis between the roots \(-2 < x < 3\)

Answer\(-2 < x < 3\)

True or False?

Decide whether each statement is true or false, and correct it if it is false. (a) \(-3 > -2\) (b) \(x \ge 5\) includes 5 (c) if \(-x < 4\) then \(x < -4\) (d) the integers satisfying \(1 \le x < 4\) are 1, 2, 3, 4.

1. Use a number line.

2. Test values.

A good answer shows: (a) False: \(-3 < -2\). (b) True. (c) False: reversing the sign gives \(x > -4\). (d) False: 4 is not included, so the integers are 1, 2, 3.

Can I...?

  1. 1Use the four inequality symbols.
  2. 2Draw an inequality on a number line.
  3. 3Solve a linear inequality.
  4. 4Reverse the sign when multiplying by a negative.
  5. 5Solve a double inequality.
  6. 6List integer solutions.
  7. 7Shade a region on a graph (Higher).
  8. 8Solve a quadratic inequality (Higher).

Summary & Exam Focus

  • Open circle for \(<\) and \(>\); filled circle for \(\le\) and \(\ge\).
  • Solve like an equation, but reverse the sign when dividing by a negative.
  • Double inequality: do the same to all three parts.
  • Higher: solid boundary for \(\le\), \(\ge\); dashed for \(<\), \(>\).

Exam focus

Solve \(5 - 2x > 11\). (3 marks) (3 marks)

Dividing by a negative number flips the sign. Check your answer with a test value.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Inequality
A statement that compares two values using \(<\), \(>\), \(\le\) or \(\ge\).
Integer
A whole number, positive, negative or zero.
Solution set
All the values that make an inequality true.
Boundary line
The line that separates a region on a graph.
Region
An area of a graph that satisfies one or more inequalities.
Strict inequality
One with \(<\) or \(>\), not including the boundary.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Non-calculator 2 marks

    Solve \(3x + 2 \le 14\).

    Show answerHide answer

    Model answer

    \(3x \le 12\), so \(x \le 4\).

    Mark scheme

    • \(3x \le 12\) — M1
    • \(x \le 4\) — A1
  2. Question 2 Non-calculator 3 marks

    Solve \(5 - 2x > 11\).

    Show answerHide answer

    Model answer

    \(-2x > 6\), so \(x < -3\).

    Mark scheme

    • \(-2x > 6\) — M1
    • Dividing by \(-2\) — M1
    • \(x < -3\) — A1
  3. Question 3 Non-calculator 2 marks

    The number line shows an inequality. Write down the inequality.

    A number line with an open circle at minus 2, a filled circle at 3 and a bar between them.
    Show answerHide answer

    Model answer

    \(-2 < x \le 3\).

    Mark scheme

    • \(x \le 3\) or \(x > -2\) — B1
    • \(-2 < x \le 3\) — B1
  4. Question 4 Non-calculator 2 marks

    \(n\) is an integer. Write down all the possible values of \(n\) when \(-2 < n \le 3\).

    Show answerHide answer

    Model answer

    \(-1, 0, 1, 2, 3\).

    Mark scheme

    • At least four correct values — M1
    • All correct — A1
  5. Question 5 Non-calculator 3 marks

    Solve \(5 < 2x - 1 \le 9\).

    Show answerHide answer

    Model answer

    \(6 < 2x \le 10\), so \(3 < x \le 5\).

    Mark scheme

    • Adding 1 to all parts — M1
    • Dividing all parts by 2 — M1
    • \(3 < x \le 5\) — A1
  6. Question 6 Non-calculator 4 marks

    On the grid, the region R satisfies \(y \ge 1\), \(y \le x\) and \(x + y \le 6\). Mark the region R with an R.

    A blank coordinate grid with x and y from -1 to 6.
    Show answerHide answer

    Model answer

    The lines \(y = 1\), \(y = x\) and \(x + y = 6\) are drawn; the triangle with vertices \((1, 1)\), \((5, 1)\) and \((3, 3)\) is labelled R.

    Mark scheme

    • One line drawn correctly — M1
    • All three lines drawn correctly — M1
    • The correct region — M1
    • R labelled — A1

Quick check

  1. Which symbol means "less than or equal to"?

    1. A\(<\)
    2. B\(\le\)
    3. C\(>\)
    4. D\(\ge\)
    Show answerHide answer

    B: \(\le\)

    \(\le\) means less than or equal to.

  2. Solve \(x + 5 > 8\).

    1. A\(x > 13\)
    2. B\(x < 3\)
    3. C\(x > 3\)
    4. D\(x = 3\)
    Show answerHide answer

    C: \(x > 3\)

    Subtract 5: \(x > 3\).

  3. Solve \(-3x < 12\).

    1. A\(x < -4\)
    2. B\(x < 4\)
    3. C\(x > 4\)
    4. D\(x > -4\)
    Show answerHide answer

    D: \(x > -4\)

    Divide by \(-3\) and reverse the sign: \(x > -4\).

  4. Which integers satisfy \(1 \le x < 4\)?

    1. A1, 2, 3
    2. B2, 3, 4
    3. C1, 2, 3, 4
    4. D2, 3
    Show answerHide answer

    A: 1, 2, 3

    1 is included and 4 is not: 1, 2, 3.

  5. On a number line, \(x \ge 2\) is shown with...

    1. AAn open circle and an arrow left
    2. BA filled circle and an arrow right
    3. CAn open circle and an arrow right
    4. DA filled circle and an arrow left
    Show answerHide answer

    B: A filled circle and an arrow right

    A filled circle at 2 (included) and an arrow to the right.

  6. Solve \(x^2 < 9\).

    1. A\(x < 3\)
    2. B\(x < -3\) or \(x > 3\)
    3. C\(-3 < x < 3\)
    4. D\(x > -3\)
    Show answerHide answer

    C: \(-3 < x < 3\)

    \(x\) lies between the roots: \(-3 < x < 3\).

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