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Cambridge IGCSE0580

Inequalities

Mathematics 0580 Chapter Notes

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Inequalities
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In this chapter

  1. 1Inequality signs and integer values
  2. 2Inequalities on a number line
  3. 3Solving linear inequalities
  4. 4Double inequalities
  5. 5Constructing an inequality
  6. 6Inequalities in two variables on a graph
  7. 7Inequalities that define a region

1. Inequality signs and integer values

An inequality compares two values. < means less than and > means greater than. These are strict inequalities: the end value is not included.

≤ means less than or equal to and ≥ means greater than or equal to. These are inclusive inequalities: the end value is included.

An integer is a whole number. To list the integers that satisfy an inequality, check each end value against its sign.

Examiner insight

“Some responses showed incorrect interpretation of the inequality symbols and were missing the –3 or including the 3. Some responses missed the zero and others did not give integer values for x.”

Worked example 12 marks

Find the integer values of x that satisfy -3 ≤ x < 1.

  1. 1

    -3 ≤ x includes -3, so the list starts at -3 [1]

  2. 2

    x < 1 leaves out 1, so the list is -3, -2, -1, 0 [1]

2. Inequalities on a number line

On a number line, an open circle marks a strict inequality (< or >). A closed circle marks an inclusive inequality (≤ or ≥). A line joins the circles to show the values between.

To write an inequality from a number line, read each end on its own. Then join them with the letter in the middle, with the smaller number on the left.

-3 ≤ x < 1

Examiner insight

“A wide variety of wrong answers were seen, including dots or vertical lines as end points (rather than circles), circles shaded the wrong way round, a line drawn with no circles and jumps drawn instead of a line.” “A significant number scored zero due to common errors such as using incorrect inequality symbols, placing the inequality between numbers (e.g., –4 < 3, –4 > 3) or simply listing –4 or 3 without forming a complete inequality statement.”

Worked example 12 marks

A number line has a closed circle at -2, an open circle at 3 and a line joining them. Write down the inequality it represents.

  1. 1

    closed circle at -2: included, so -2 ≤ x [1]

  2. 2

    open circle at 3: not included, so -2 ≤ x < 3 [1]

3. Solving linear inequalities

Solve a linear inequality as you would an equation: do the same to both sides. Collect the terms in x on one side and the numbers on the other.

There is one difference. When you multiply or divide both sides by a negative number, reverse the inequality sign.

Give the answer as an inequality, such as x < 3.5. An answer of x = 3.5 or 3.5 alone loses the mark.

-2x > 6 ⇒ x < -3

Examiner insight

“The two main errors were that the sign was not changed when dividing both sides by –2, e.g. –2x ⩾ –6 leading to x ⩾ 3 and a sign error e.g. 2x ⩽ –6 and therefore x ⩽ –3 is written after the inequality is rearranged.” “The most successful candidates rearranged to keep the terms in n positive i.e. 18 –11 > 5n – 3n.”

Worked example 13 marks

Do not use a calculator. Solve 5 - 2x > 11.

  1. 1

    subtract 5 from both sides: -2x > 6 [1]

  2. 2

    divide both sides by -2 and reverse the sign [1]

  3. 3

    x < -3 [1]

4. Double inequalities

A double inequality such as -3 ≤ 3x - 2 < 7 has three parts. Whatever you do, do it to all three parts.

The signs stay in the same direction unless you multiply or divide by a negative number.

If the question asks for integer values, list the whole numbers in the final range. Check both ends against their signs. Zero is not a positive integer.

Examiner insight

“Sometimes they were then unable to find the integer values or wrongly included 0 as a positive integer.”

Worked example 13 marks

Do not use a calculator. Find the positive integer values of x that satisfy -3 ≤ 3x - 2 < 7.

  1. 1

    add 2 to every part: -1 ≤ 3x < 9 [1]

  2. 2

    divide every part by 3: -1/3 ≤ x < 3 [1]

  3. 3

    the integers in the range are 0, 1, 2 and 0 is not positive, so x = 1, 2 [1]

5. Constructing an inequality

To construct an inequality from words, first write an expression for the quantity. Write an inequality, not an equation.

Then choose the sign. If the limit itself is allowed, use an inclusive sign (≤ or ≥). If it is not allowed, use a strict sign (< or >).

Solve it, then give the answer as an inequality, such as n > 25, not as 25 alone.

Examiner insight

“Having solved the inequality correctly some gave an answer of 25 rather than n > 25. A significant number of candidates used an incorrect inequality symbol, either < or ⩾ and some set up an equation rather than an inequality.”

Worked example 13 marks

A taxi costs £4 plus £2 for each kilometre. Ravi has £30. Write an inequality for the distance d km he can travel and solve it.

  1. 1

    cost must not be more than 30: 4 + 2d ≤ 30 [1]

  2. 2

    subtract 4: 2d ≤ 26 [1]

  3. 3

    d ≤ 13 [1]

6. Inequalities in two variables on a graph

Draw the boundary line first, using the equals sign. Use a broken line for a strict inequality (<, >). Use a solid line for an inclusive one (≤, ≥).

Shade the unwanted region, unless the question says otherwise. Label the wanted region as the question asks, such as R.

To decide which side is wanted, test a point not on the line. Substitute it into the inequality.

y > 2x - 1: test (0,0), 0 > -1 is true

Examiner insight

“Strict inequalities, such as y < 10, should use broken lines and inclusive inequalities, such as x ⩽ 16, should use solid lines.” “Many did not label their intended region with R and some seemed to be shading the wanted rather than unwanted regions as instructed in the question.”

Worked example 12 marks

The region y > 2x - 1 is wanted. Show whether the point (2, 4) lies in the wanted region and state the type of boundary line.

  1. 1

    substitute (2, 4): 4 > 2 × 2 - 1 = 3 is true, so the point is in the wanted region [1]

  2. 2

    > is strict, so the line y = 2x - 1 is broken [1]

7. Inequalities that define a region

To list the inequalities for a region, take each boundary line in turn. Find its equation. A solid line gives ≤ or ≥. A broken line gives < or >.

Then choose the direction of the sign. Pick a point inside the region and check it satisfies each inequality.

A region needs one inequality for each boundary line.

Examiner insight

“Candidates should be encouraged to choose a point within the region to check whether their inequality sign is the correct way round.” “Many candidates made an error with one or more of the inequalities either by reversing the direction of the inequality or by including/excluding equality inappropriately.”

Worked example 13 marks

Region R is bounded by three lines: the solid line x = 2, the broken line y = 5 and the solid line y = x. The point (3, 4) lies inside R. Write down the three inequalities that define R.

  1. 1

    x = 2 is solid and 3 > 2: x ≥ 2 [1]

  2. 2

    y = 5 is broken and 4 < 5: y < 5 [1]

  3. 3

    y = x is solid and 4 > 3: y ≥ x [1]

Recap

  • < and > leave out the end value. ≤ and ≥ include it.
  • Number line: open circle for strict, closed circle for inclusive.
  • Extended: solve as an equation, but reverse the sign when multiplying or dividing by a negative.
  • Extended: in a double inequality such as -3 ≤ 3x - 2 < 7, do the same to all three parts.
  • Extended: broken line for strict, solid line for inclusive. Shade the unwanted region.
  • Extended: to list the inequalities for a region, find each line and test a point inside.

End-of-chapter exercise

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

  1. Write "x is at least -2" using an inequality sign.1 mark
  2. List the integer values of x that satisfy -2 < x ≤ 3.2 marks
  3. A number line has an open circle at -1, a closed circle at 4 and a line joining them. Write down the inequality it represents.2 marks
  4. Do not use a calculator. Find the largest integer n with n < 7.5 and the smallest integer m with m ≥ -2.5.2 marks
  5. Do not use a calculator. Solve 4x - 3 > 9.2 marks
  6. Do not use a calculator. Solve 7 - 3x ≥ 22.3 marks
  7. The inequality x + 2y ≤ 8 is drawn on a grid with the unwanted region shaded. (a) State whether the line x + 2y = 8 is solid or broken. (b) Decide whether each of the points (2, 3) and (4, 3) is in the wanted region.3 marks
  8. Do not use a calculator. Find the integer values of x that satisfy -5 ≤ 2x + 1 < 9.3 marks
  9. Do not use a calculator. Two candles are lit at the same time. Candle A is 30 cm tall and burns down 2.5 cm each hour. Candle B is 21 cm tall and burns down 1 cm each hour. (a) Write an inequality in t for the times, t hours after lighting, when candle A is taller than candle B. Solve it. (b) The heights are measured at the end of each whole hour, starting at t = 1. Find the number of measurements at which candle A is taller than candle B.5 marks
  10. Region R is bounded by the broken line y = 2x, the solid line x + y = 6 and the solid line y = 1. The point (2, 2) lies inside R. (a) Write down the three inequalities that define R. (b) Find the number of points in R whose coordinates are both integers.6 marks

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