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Linear Inequalities And Shading

Linear Inequalities And Shading

Solving linear inequalities

Linear Inequalities Core + Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. Symbols and number lines
  4. Solving an inequality
  5. Double inequalities and integers
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 2.6: inequalities
  • 2.6Use \(<,\ >,\ \le,\ \ge\) and show an inequality on a number line.
  • 2.6Solve linear inequalities, including double inequalities.
  • 2.6List the integers that satisfy an inequality.

Before you start: Linear equations. Next: Regions and shading.

Formula bank

The one new ruleLearn itmultiply or divide by a NEGATIVE → reverse the sign\(-2x>8\Rightarrow x<-4\)Use it when the number in front of x is negative.
Number lineLearn it\(<,\ >\): open circle; \(\le,\ \ge\): filled circleopen = not included, filled = included.Use it when drawing or reading a number line.

Symbols and number lines

InequalityMeaningNumber line
\(x>2\)more than 2 (2 not included)open circle at 2, arrow right
\(x\ge2\)2 or morefilled circle at 2, arrow right
\(-1<x\le3\)between −1 and 3, 3 includedopen at −1, filled at 3, line between
−1 < x ≤ 3-3-2-1012345
\(-1<x\le3\): open circle = not included, filled circle = included.

Solving an inequality

Solve it exactly like an equation. The only difference: multiplying or dividing by a negative number reverses the sign, because \(2<5\) but \(-2>-5\).

General methodHow to solve a linear inequality

  1. Expand brackets; clear fractions by multiplying by a POSITIVE number.
  2. Collect x on one side and numbers on the other (put x on the side where it stays positive, then no reversing is needed).
  3. Divide by the number in front of x. If it is negative, reverse the sign.
  4. Keep the inequality sign in the answer (not '=').
  5. Check with a value inside your answer.

Example Solve \(7-2x>15\).

Subtract 7, then divide by −2 and reverse the sign.

  1. \(-2x>8\)
  2. \(x<-4\)
  3. check \(x=-5\): \(7+10=17>15\) ✓

Answer \(x<-4\)

Try it yourself

1
Solve \(2x+1>9\).
Show answer

Answer \(x>4\)

2
Solve \(4-x\le1\).
Hint
Divide by −1: reverse the sign.
Show answer

Answer \(x\ge3\)

3
Solve \(4x+3>2x-9\).
Show answer

Answer \(x>-6\)

Double inequalities and integers

General methodHow to solve a double inequality and list integers

  1. Do the same thing to ALL THREE parts.
  2. End with x alone in the middle.
  3. List the whole numbers in the range: include an end only if it has ≤ or ≥.

Example Solve \(-4<2x+2\le8\) and list the integers.

Subtract 2 from all three parts, then halve.

  1. \(-6<2x\le6\)
  2. \(-3<x\le3\)

Answer −2, −1, 0, 1, 2, 3

Try it yourself

1
List the integers with \(1<2x-3\le9\).
Show answer

Answer 3, 4, 5, 6

2
List the integers with \(-3\le2x-5<7\).
Show answer

Answer 1, 2, 3, 4, 5

Ways they test it

Type 1 Solve

"Solve \(5-3x\le2(x-5)\)." (2 marks)

Spot it one inequality sign, x on one or both sides.

  1. Expand.
  2. Collect x where it is positive.
  3. Divide.

For example \(5-3x\le2x-10\)

  1. \(15\le5x\)

Answer \(x\ge3\)

Type 2 Read a number line

"Write down the inequality shown." (1 mark)

Spot it a diagram with circles.

  1. Open circle: < or >. Filled: ≤ or ≥.
  2. Two circles: a double inequality.

Type 3 List integers

"List the integers that satisfy \(-1\le3x-1<11\)." (2 marks)

Spot it 'integers' or 'whole numbers'.

  1. Solve all three parts.
  2. List, checking each end.

For example \(0\le x<4\)

    Answer 0, 1, 2, 3

    Type 4 Positive integers

    "Find the positive integers n with \(3n-8<n+5\)."

    Spot it 'positive' cuts off at 1.

    1. Solve.
    2. List from 1 upward.

    For example \(2n<13\Rightarrow n<6.5\)

      Answer 1, 2, 3, 4, 5, 6

      Mistakes to avoid

      • ✗ \(-2x>8\Rightarrow x>-4\)✓ \(x<-4\)Dividing by a negative reverses the sign.
      • ✗ \(x=5\) as the answer✓ \(x<5\)Keep the inequality.
      • ✗ Including the end of a < in a list✓ leave it out< means not included.
      • ✗ Filled circle for <✓ open circleNot included.

      Summary

      • Solve like an equation.
      • × or ÷ by a negative → reverse the sign.
      • Double inequality: the same operation on all three parts.
      • Open circle for < >, filled for ≤ ≥.

      If you see …

      • a negative number in front of xmove x to the other side, or reverse when you divide.
      • "list the integers"check whether each end is included.

      Practice

      Practice: easy, medium, hard

      Easy

      1
      Solve \(3x-5<10\).
      Show answer

      Answer \(x<5\)

      2
      Solve \(-2x+7\ge1\).
      Hint
      Divide by −2: reverse.
      Show answer

      Answer \(x\le3\)

      3
      List the integers with \(-4<2x+2\le8\).
      Show answer

      Answer −2, −1, 0, 1, 2, 3

      Medium

      1
      Solve \(5-3x\le2(x-5)\).
      Guide me

      What is it asking? x

      What do we know? expand the bracket

      Plan collect x on the right

      Show working
      1. \(5-3x\le2x-10\)
      2. \(15\le5x\)
      3. \(x\ge3\)

      Answer \(x\ge3\)

      2
      Solve \(5-3x<2x+20\).
      Guide me

      What is it asking? x

      What do we know? x on both sides

      Plan add 3x, subtract 20

      Show working
      1. \(-15<5x\)
      2. \(x>-3\)

      Answer \(x>-3\)

      3
      List the integers with \(-1\le3x-1<11\).
      Guide me

      What is it asking? a list

      What do we know? add 1 to all parts

      Plan divide by 3

      Show working
      1. \(0\le3x<12\)
      2. \(0\le x<4\)

      Answer 0, 1, 2, 3

      Hard

      1
      Solve \(\dfrac x3+2\ge\dfrac{x-1}2\).
      Guide me

      What is it asking? x

      What do we know? × 6 (positive)

      Plan \(2x+12\ge3x-3\)

      Show working
      1. \(15\ge x\)
      2. \(x\le15\)

      Answer \(x\le15\)

      PartAnswer / workingMarksGuidance
      \(2x+12\ge3(x-1)\)M1
      \(15\ge x\)M1
      \(x\le15\)A1
      Total: 3 marks
      2
      Find all the positive integers n with \(3n-8<n+5\).
      Guide me

      What is it asking? a list

      What do we know? \(2n<13\)

      Plan positive: start at 1

      Show working
      1. \(n<6.5\)
      2. 1, 2, 3, 4, 5, 6

      Answer 1, 2, 3, 4, 5, 6

      PartAnswer / workingMarksGuidance
      \(n<6.5\)M1
      1, 2, 3, 4, 5, 6A1
      Total: 2 marks
      3
      Solve \(7-2x>15\).
      Guide me

      What is it asking? x

      What do we know? subtract 7

      Plan ÷ −2 and reverse

      Show working
      1. \(-2x>8\)
      2. \(x<-4\)

      Answer \(x<-4\)

      PartAnswer / workingMarksGuidance
      \(-2x>8\)M1
      \(x<-4\)A1
      Total: 2 marks

      Can I do this?

      Anything unticked: revise that section first. All ticked: take the quiz.

      Quiz

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