Double inequalities and integer solutions

The Maths Habit

1List the integers that satisfy −4 < x ≤ 0.−3, −2, −1, 02Solve 10 − 3x > 1.x < 33A mass m is 7 kg to the nearest kg, so 6.5 ≤ m < 7.5. Is 7.5 kg included? Is 6.5 kg?No; yes.

Think of a number. Double it and add 1. The answer is more than −5 and at most 13. What could your number be?

Any number greater than −3, up to and including 6.

As an inequality: −5 < 2n + 1 ≤ 13 gives −3 < n ≤ 6.

Double inequality
An inequality with two signs, which puts x between two values.

Do the same to all three parts.

Solve−5≤3x + 1<13
− 1 from all three parts−6≤3x<12
÷ 3 all three parts−2≤x<4
Answer−2 ≤ x < 4
Check: try a value from your answer in the question. Is it true?x = 0: −5 ≤ 1 < 13 is true

Do the same to all three parts.

Solve1<2x − 5≤9
+ 5 to all three parts6<2x≤14
÷ 2 all three parts3<x≤7
Answer3 < x ≤ 7
Check: try a value from your answer in the question. Is it true?x = 5: 1 < 5 ≤ 9 is true

Do the same to all three parts.

Solve−5≤3x + 1<13
− 1 from all three parts−6≤3x<12
÷ 3 all three parts−2≤x<4
Answer−2 ≤ x < 4
Check: try a value from your answer in the question. Is it true?x = 0: −5 ≤ 1 < 13 is true

Do the same to all three parts.

Solve−11<4x + 5≤13
− 5 from all three parts−16<4x≤8
÷ 4 all three parts−4<x≤2
Answer−4 < x ≤ 2
Check: try a value from your answer in the question. Is it true?x = 0: −11 < 5 ≤ 13 is true

n is an integer. 3n − 4 > 2 and 2n + 1 ≤ 13. Find all the values of n.

3n − 4 > 2→3n > 6→n > 22n + 1 ≤ 13→2n ≤ 12→n ≤ 6
Answer3, 4, 5, 6

n is an integer. 5n + 3 ≥ −7 and 3n − 2 < 4. Find all the values of n.

5n + 3 ≥ −7→5n ≥ −10→n ≥ −23n − 2 < 4→3n < 6→n < 2
Answer−2, −1, 0, 1

n is an integer. 3n − 4 > 2 and 2n + 1 ≤ 13. Find all the values of n.

3n − 4 > 2→3n > 6→n > 22n + 1 ≤ 13→2n ≤ 12→n ≤ 6
Answer3, 4, 5, 6

n is an integer. 6n + 1 > −11 and 9 − 2n ≥ 2. Find all the values of n.

6n + 1 > −11→6n > −12→n > −29 − 2n ≥ 2→−2n ≥ −7→n ≤ 3.5turn the sign round
Answer−1, 0, 1, 2, 3

Solve −6 < 3x + 6 ≤ 21.

A−4 < x ≤ 5
B−2 < x ≤ 7
C−12 < x ≤ 15
Dx ≤ 5
A: rightB: the 6 comes off all three partsC: ÷ 3 all three partsD: keep both ends: −4 < x as well

Double inequalities and integer solutions worksheet, page 1

Answer sheet

  1. A removal firm charges £45 plus £3 for each mile. A job costs more than £60 and no more than £120. Write and solve a double inequality for the number of miles, m.

    60 < 45 + 3m ≤ 120, so 5 < m ≤ 25

  2. Ali says −2 < x < 3 means the same as x > −2 and x < 3. Is he right?

    Yes. Both say x is more than −2 and less than 3. For a between answer, "and" is right.

  3. Write a double inequality, with one end strict and one not, whose integer solutions are exactly 4, 5 and 6. Then write one in the form a < 3x − 1 ≤ b.

    One of many: 3 < x ≤ 6. Then 8 < 3x − 1 ≤ 17. (3.5 ≤ x < 7 also works.)

  1. Solve −6 < 5x + 4 ≤ 19.

    −2 < x ≤ 3

  2. List the integers that satisfy your answer to Q1.

    −1, 0, 1, 2, 3

  3. n is an integer. 2n − 3 > 1 and 3n + 2 < 20. Find all the values of n.

    3, 4, 5
Exit tickets to print