Teach Me Maths

Solving Linear Inequalities

Question 11 mark

Which statement means x<6?

Hint

Check whether the symbol includes the boundary value.

Worked solution

Which statement means x<6?

  1. x is less than 6.
  2. x is at most 6.
  3. x is greater than 6.
  4. x is exactly 6.
  1. The symbol < means less than. It does not include equality, so 6 itself is not allowed.

Question 21 mark

The number line shows the possible values of n. Write down the greatest integer value of n.

Hint

A filled circle includes its endpoint. Follow the highlighted direction.

Worked solution

The number line shows the possible values of n. Write down the greatest integer value of n.

  1. The filled circle includes -1. The highlighted ray extends to smaller values, so the greatest integer is -1.

Question 31 mark

The number line shows the possible values of p. Write down the smallest integer value of p.

Hint

An open circle leaves out its endpoint.

Worked solution

The number line shows the possible values of p. Write down the smallest integer value of p.

  1. The open circle at 2 means p>2. The first integer greater than 2 is 3.

Question 41 mark

Write all the integers that satisfy -3\leq t<2.

Give every value, separated by commas

Hint

Include the lower endpoint but leave out the upper endpoint.

Worked solution

Write all the integers that satisfy -3\leq t<2.

  1. -3 is included because of \leq. The integer 2 is excluded because of <.
  2. The integers are -3, -2, -1, 0 and 1.

Question 51 mark

Which number line represents -2<n\leq2?

Hint

Check each endpoint separately, then check the highlighted interval.

Worked solution

Which number line represents -2<n\leq2?

  1. A
  2. B
  3. C
  4. D
  1. -2 is excluded, so its circle must be open. The value 2 is included, so its circle must be filled.
  2. B has these endpoints and highlights the values between them.

Question 61 mark

Solve x+7<12.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Subtract the same number from both sides.

Worked solution

Solve x+7<12.

  1. Subtract 7 from both sides: x<12-7.
  2. So x<5.

Question 71 mark

Solve 4y\geq28.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Divide both sides by the positive multiplier.

Worked solution

Solve 4y\geq28.

  1. Divide by 4: y\geq28\div4.
  2. The divisor is positive, so the sign stays the same: y\geq7.

Question 82 marks

Solve 3x+8\leq-1.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Undo the addition before the multiplication.

Worked solution

Solve 3x+8\leq-1.

  1. Subtract 8: 3x\leq-9.
  2. Divide by 3: x\leq-3.

Question 91 mark

Zara says this number line represents -1\leq x<3. What is wrong?

Hint

Compare the two inequality signs with the circles drawn.

Worked solution

Zara says this number line represents -1\leq x<3. What is wrong?

  1. The circle at −1 should be filled.
  2. The circle at 3 should be filled.
  3. The line should continue to the left.
  4. Both circles should be filled.
  1. -1\leq x includes -1, so the open circle there must be filled.
  2. x<3 excludes 3, so the open circle at 3 is correct.

Question 102 marks

Find the greatest integer n that satisfies 2n+3<14.

Hint

Solve the inequality first, then choose an integer strictly inside the range.

Worked solution

Find the greatest integer n that satisfies 2n+3<14.

  1. 2n<11, so n<5.5.
  2. The greatest integer below 5.5 is 5.

Question 112 marks

A notebook costs £3. Mia has £14 and must buy at least two notebooks. Write all the possible whole numbers of notebooks she can buy.

Give every value, separated by commas

Hint

The total cost cannot exceed the money available.

Worked solution

A notebook costs £3. Mia has £14 and must buy at least two notebooks. Write all the possible whole numbers of notebooks she can buy.

  1. If n notebooks are bought, 3n\leq14, so n\leq14\div3=4\frac23.
  2. At least two are needed, so the possible whole numbers are 2, 3 and 4.

Question 122 marks

Solve 2(x+4)>18.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

You can divide both sides by 2 before removing the bracket.

Worked solution

Solve 2(x+4)>18.

  1. Divide by 2: x+4>9.
  2. Subtract 4: x>5.

Question 133 marks

Solve 5x-7\leq2x+8.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Collect the x-terms on one side and the constants on the other.

Worked solution

Solve 5x-7\leq2x+8.

  1. Subtract 2x: 3x-7\leq8.
  2. Add 7: 3x\leq15. Divide by 3: x\leq5.

Question 142 marks

A lift can carry at most 430 kg. A trolley weighs 70 kg and each crate weighs 60 kg. At least three crates must go in the lift. Write all the possible numbers of crates.

Give every value, separated by commas

Hint

Include the trolley mass before applying the lift limit.

Worked solution

A lift can carry at most 430 kg. A trolley weighs 70 kg and each crate weighs 60 kg. At least three crates must go in the lift. Write all the possible numbers of crates.

  1. For n crates, 70+60n\leq430. This gives 60n\leq360, so n\leq6.
  2. With at least three crates, the possible numbers are 3, 4, 5 and 6.

Question 152 marks

The number line shows the solution of 2x+k>11, where k is a constant. Find k.

Hint

The endpoint gives the boundary where the two sides would be equal.

Worked solution

The number line shows the solution of 2x+k>11, where k is a constant. Find k.

  1. The number line shows x>3. Solving the given inequality gives x>\frac{11-k}{2}.
  2. Therefore \frac{11-k}{2}=3. So 11-k=6 and k=5.

Question 16Challenge3 marks

Solve 4(x-3)\geq13-x.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Expand the bracket first, then collect the x-terms on one side of the inequality.

Worked solution

Solve 4(x-3)\geq13-x.

  1. Expand the bracket: 4x-12\geq13-x.
  2. Add x to both sides: 5x-12\geq13.
  3. Add 12 to both sides: 5x\geq25.
  4. Divide both sides by 5: x\geq5.

Question 17Challenge3 marks

Find the least integer n that satisfies \frac{3n-2}{4}>5.

Hint

Remove the denominator, then find the first integer above the boundary.

Worked solution

Find the least integer n that satisfies \frac{3n-2}{4}>5.

  1. Multiply by 4: 3n-2>20. Add 2: 3n>22.
  2. Divide by 3: n>\frac{22}{3}=7\frac13.
  3. The least integer allowed is 8.

Question 18Challenge3 marks

The rectangle has perimeter at most 49 cm. The number x is a whole number and x>1. Find the greatest possible value of x.

Hint

Form the perimeter using both pairs of equal sides.

Worked solution

The rectangle has perimeter at most 49 cm. The number x is a whole number and x>1. Find the greatest possible value of x.

  1. The perimeter is 2(2x+1)+2(x-1)=4x+2+2x-2=6x cm.
  2. 6x\leq49, so x\leq\frac{49}{6}=8\frac16.
  3. The greatest whole number is 8. It gives side lengths 17 cm and 7 cm, with perimeter 48 cm.

Question 19Challenge3 marks

Write all the integers that satisfy -5<3x+1\leq13.

Give every value, separated by commas

Hint

Apply each operation to all three parts of the inequality.

Worked solution

Write all the integers that satisfy -5<3x+1\leq13.

  1. Subtract 1 throughout: -6<3x\leq12.
  2. Divide throughout by 3: -2<x\leq4.
  3. The integers are -1, 0, 1, 2, 3 and 4.

Question 20Challenge4 marks

For a delivery of n parcels, company A charges £8 plus £1.50 per parcel. Company B charges £2 plus £2.50 per parcel.

A must be cheaper than B, and the charge from A must be at most £23. Write all the possible positive whole-number values of n.

Give every value, separated by commas

Hint

There are two conditions. Find the values allowed by each, then keep those allowed by both.

Worked solution

For a delivery of n parcels, company A charges £8 plus £1.50 per parcel. Company B charges £2 plus £2.50 per parcel.

A must be cheaper than B, and the charge from A must be at most £23. Write all the possible positive whole-number values of n.

  1. A is cheaper when 8+1.5n<2+2.5n. Subtract 1.5n and 2 to get n>6.
  2. The budget gives 8+1.5n\leq23, so 1.5n\leq15 and n\leq10.
  3. Together, 6<n\leq10. The positive whole numbers are 7, 8, 9 and 10.

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