Skip to content
Cambridge IGCSE0607

Inequalities

International Mathematics 0607 Chapter Notes

What this chapter covers

Inequalities
ShareWhatsAppPost
Inequalities notes

Read online free. Download a watermarked copy with a free account.

Read the notes

The full Inequalities notes as text: skim, search, and jump between subtopics.

~8 min read

In this chapter

  1. 1Inequality signs and the number line
  2. 2Integers that satisfy an inequality
  3. 3Solving a linear inequality
  4. 4Constructing an inequality from a problem
  5. 5Solving an inequality with a graphic display calculator
  6. 6Showing an inequality in two variables on a graph
  7. 7Listing the inequalities that define a region

1. Inequality signs and the number line

An inequality compares two quantities using <, >, leqslant or geqslant. The signs < and > are strict: the end value is not allowed. The signs leqslant and geqslant are inclusive: the end value is allowed.

On a number line, use an open circle for a strict inequality and a closed circle for an inclusive one. Join two circles with a continuous line. A one-sided inequality has one circle and an arrow in the direction of the values.

Examiner insight

“A few did not fill in the circle at 4, some drew the arrow in the wrong direction and others just put a cross at 4 or circled the diagram from 4 to 6.” “There were a few candidates who did not recognise that the filled in circle is different to the open circle so repeated the given less than sign when it should have been less than or equal.”

Worked example 12 marks

On the number line, show the inequality -3 leqslant x < 2.

  1. 1

    leqslant is inclusive: closed circle at -3. < is strict: open circle at 2 [1]

  2. 2

    a continuous line joining the two circles [1]

2. Integers that satisfy an inequality

A double inequality gives a range of values. Check each end against its sign before you list anything: a strict sign leaves the end value out, an inclusive sign keeps it.

When the question asks for integers, list the whole numbers in the range. Do not give the interval.

Examiner insight

“Of those who gave integers as their answer, a few included the 7 as well as the correct integers.” “In part (ii), ‘integer’ was a stumbling block. An answer of 1.9 was common.”

Worked example 12 marks

Write down all the integer values of n that satisfy this inequality.

2 < n leqslant 6

  1. 1

    n is more than 2, so 2 is not included. n can equal 6, so 6 is included [1]

  2. 2

    n = 3, 4, 5, 6 [1]

3. Solving a linear inequality

Solve a linear inequality as you solve an equation: collect the x terms on one side and the numbers on the other. One rule is new. When you multiply or divide both sides by a negative number, you must reverse the inequality sign.

In a double inequality, do each step to all three parts. Give the answer as an inequality, not as a single value.

3x < 2x + 4 ⇒ x < 4

-3 leqslant 3x - 2 < 7 ⇒ -1 leqslant 3x < 9 ⇒ -1/3 leqslant x < 3

Examiner insight

“Most candidates were confused by the combination of a negative sign and an inequality sign and therefore obtained 2.5 but with an incorrect inequality sign for their final answer.” “Some reversed the sign at the final stage when dividing by 3.”

Worked example 12 marks

Solve the inequality.

17 + 2n < 5n - 10

  1. 1

    collect terms: 17 + 10 < 5n - 2n, so 27 < 3n [1]

  2. 2

    divide by 3 (positive, so the sign stays): n > 9 [1]

4. Constructing an inequality from a problem

Let a letter stand for the unknown quantity. Write the cost, length or total in terms of that letter. Then compare it with the limit the problem gives, using the sign that fits the words.

The question may say to write down and solve an inequality. An equation does not answer it. Write the inequality, solve it, and leave the answer as an inequality such as x leqslant 4.

Examiner insight

“A significant number of candidates gave an equation, rather than an inequality, correctly found the value of 4, and then gave an answer of 4 hours.”

Worked example 13 marks

The cost to hire a boat is textdollar2 per hour plus a fixed fee of textdollar7. Mosna has textdollar15. Mosna hires a boat for x hours. Write down and solve an inequality for the number of hours that Mosna can hire a boat.

  1. 1

    cost is at most textdollar15: 2x + 7 leqslant 15 [1]

  2. 2

    subtract 7: 2x leqslant 8 [1]

  3. 3

    x leqslant 4 [1]

5. Solving an inequality with a graphic display calculator

Enter each side of the inequality as a function and draw both graphs. For an inequality compared with zero, enter the single function. Read off the x-values where the graphs meet, or where the curve crosses the x-axis.

Then use the sketch to decide which part of the x-axis satisfies the sign. For x² + 3x - 6 > 0 the curve is above the axis outside its two roots, so x < -4.37 or x > 1.37. Use the sketch, not algebra.

Examiner insight

“A number of candidates plotted points rather than using the graphic display calculator.” “Many candidates tried to use an algebraic approach rather than use their sketch to solve the inequality. This was almost always unsuccessful.”

Worked example 12 marks

Solve the inequality 2^(x) > 0.5x + 1.5.

  1. 1

    graph y = 2^(x) and y = 0.5x + 1.5: they meet at x = -2.69 and x = 1

  2. 2

    2^(x) is above the line to the right of the second meeting point: x > 1 [1]

  3. 3

    and to the left of the first: x < -2.69 [1]

6. Showing an inequality in two variables on a graph

To show an inequality such as x + y < 6, first draw its boundary line x + y = 6. Use a broken line for a strict inequality (<, >) and a solid line for an inclusive one (leqslant, geqslant).

Then shade the unwanted region, unless the question says otherwise. The wanted region is the part left clear. If you shade the wanted region instead, add a key.

Examiner insight

“There were some candidates who gave different types of shading with a hint of a valid region but no key or indication as to which was showing the candidate’s answer.”

Worked example 14 marks

Show the region where x geqslant 1, y geqslant 1 and x + y < 6. Shade the unwanted regions and label the region that is left R.

  1. 1

    x = 1 as a solid line (inclusive). Shade the side where x < 1 [1]

  2. 2

    y = 1 as a solid line (inclusive). Shade the side where y < 1 [1]

  3. 3

    x + y = 6 through (6, 0) and (0, 6) as a broken line (strict). Shade the side where x + y > 6 [1]

  4. 4

    R is the clear triangle with vertices (1, 1), (5, 1) and (1, 5) [1]

7. Listing the inequalities that define a region

Work backwards from the region. Each boundary line gives one inequality, so count the boundaries and write one for each.

For each line, find its equation. A solid line gives leqslant or geqslant and a broken line gives < or >. Then decide which side of the line the region lies on.

Examiner insight

“Many obtained part marks for drawing lines for the boundaries of their inequalities. Many did not consider all five inequalities when defining their region.”

Worked example 13 marks

The region R is the triangle with vertices (1, 1), (5, 1) and (5, 5). The side from (5, 1) to (5, 5) is a broken line. The other two sides are solid lines. Write down the three inequalities that define R.

  1. 1

    side from (1, 1) to (5, 1) is y = 1, solid, with R above it: y geqslant 1 [1]

  2. 2

    side from (1, 1) to (5, 5) is y = x, solid. (5, 1) is below it: y leqslant x [1]

  3. 3

    side from (5, 1) to (5, 5) is x = 5, broken, with R to the left: x < 5 [1]

Recap

  • Strict signs (<, >): open circle on a number line, broken line on a graph.
  • Inclusive signs (leqslant, geqslant): closed circle on a number line, solid line on a graph.
  • List integers by checking each end against its sign.
  • Solve as an equation, but reverse the sign when multiplying or dividing by a negative number.
  • Construct an inequality from the problem, solve it, and give an inequality as the answer.
  • On a graphic display calculator, sketch the graphs and read the sketch to choose the signs.
  • Shade the unwanted region. Write one inequality for each boundary of a region.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Write x is at least 3 and less than 9 as an inequality.2 marks
  2. Do not use a calculator. Write down all the integers n that satisfy -3 < n leqslant 2.2 marks
  3. Describe how to show -2 < x leqslant 5 on a number line.2 marks
  4. Do not use a calculator. Solve 7 - 2x < 15.2 marks
  5. Do not use a calculator. Solve -7 leqslant 4x - 1 < 9.3 marks
  6. A taxi costs textdollar3 plus textdollar1.50 for each kilometre. Ali has textdollar24. Write down and solve an inequality for the number of kilometres, k, that Ali can travel.3 marks
  7. Use a graphic display calculator to solve x² - 2x - 5 leqslant 0. Give your values correct to 3 significant figures.3 marks
  8. Do not use a calculator. The region R is the triangle with vertices (0, 0), (6, 0) and (0, 3). The side from (6, 0) to (0, 3) is a broken line. The other two sides are solid lines. Write down the three inequalities that define R.4 marks
  9. Do not use a calculator. A candle is 30 cm tall when it is lit. Its height decreases by 2.5 cm every hour. After t hours its height is more than 8 cm and at most 20 cm. (a) Write down an inequality in t and solve it. (b) Write down all the integer values of t.5 marks
  10. A baker makes x trays of bread and y trays of cakes each day. He makes at least 2 trays of bread and fewer than 6 trays of cakes. He makes more trays of cakes than trays of bread. He makes at most 8 trays altogether. (a) Write down the four inequalities that define the region of possible points (x, y). (b) Write down the equations of the boundary lines that are drawn as broken lines. (c) x and y are integers. Find the greatest possible value of x and the values of y that go with it.6 marks

Free notes are just the start.

Unlock every Workbook and Chapter at a Glance, and generate your own worksheets and predicted papers.

Explore plans

Related chapters

Types of numberSetsPowers and rootsFractions, decimals and percentagesOrderingThe four operations