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Either paper 1 mark
\(x\) is no more than \(-1\) and greater than or equal to \(-5\).
Circle the inequality that shows this.
A: \(-5 ≤ x < -1\)
B: \(-5 < x < -1\)
C: \(-5 ≤ x ≤ -1\)
D: \(-5 < x ≤ -1\)
Show answer
C: \(-5 ≤ x ≤ -1\)"greater than or equal to \(-5\)" gives the lower end: the sign next to \(-5\) is \(≤\), and \(<\) leaves \(-5\) out while \(≤\) includes it.
"no more than \(-1\)" gives the upper end: the sign next to \(-1\) is \(≤\).
Write the smaller number, \(-5\), on the left and the larger number, \(-1\), on the right:
\(-5 ≤ x ≤ -1\)
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Either paper 1 mark
Circle the list of all the integers that satisfy
\(-3 < x ≤ 2\)
A: \(-2,\ -1,\ 0,\ 1,\ 2\)
B: \(-3,\ -2,\ -1,\ 0,\ 1,\ 2\)
C: \(-2,\ -1,\ 0,\ 1\)
D: \(-3,\ -2,\ -1,\ 0,\ 1\)
Show answer
\(-2,\ -1,\ 0,\ 1,\ 2\)A sign ≤ keeps the end value, and a sign \(<\) drops it.
Lower end: \(<\) so -3 is not included. Upper end: \(≤\) so 2 is included.
The integers run from -2 to 2, with every whole number in between:
\(-2,\ -1,\ 0,\ 1,\ 2\)
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Either paper 2 marks
Here is an inequality.
Write down the inequality shown.
Show answer
\(x > 2\)The circle at \(2\) is open, so \(2\) itself is not included.
The shading goes to the right, so every value bigger than \(2\) is in the solution.
The inequality is \(x > 2\)
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Either paper 3 marks
Here is an inequality.
(a)Write down the inequality shown.
(b)Write down the largest integer that satisfies this inequality.
Show answer
(a) \(x \leq -1\)
(b) \(-1\)The circle at \(-1\) is filled in, so \(-1\) itself is included.
The shading goes to the left, so every value up to \(-1\) is in the solution.
The inequality is \(x \leq -1\)
A filled circle means \(-1\) is allowed, so it is the largest value itself.
The largest integer is \(-1\).
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Either paper 1 mark
\(n\) is an integer.
The largest possible value of \(n\) is \(0\).
Circle the correct inequality.
\(n > 0\) \qquad \(n < 0\) \qquad \(n ≤ 0\) \qquad \(n ≥ 0\)
Show answer
\(n ≤ 0\)The largest possible value of \(n\) is \(0\), so \(n\) can equal \(0\) and the sign includes it.
Every other value of \(n\) lies below \(0\), so the inequality is \(n ≤ 0\).
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Either paper 3 marks
\(n\) is an integer and \(-2 < n < 3\).
Write down all the possible values of \(n\).
Show answer
\(-1, 0, 1, 2\)Both circles are open, so \(-2\) and \(3\) are not included.
Count up from \(-2\), stopping before \(3\):
\(n = -1, 0, 1, 2\)
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Either paper 3 marks
\(a\) and \(b\) are whole numbers.
\(2 ≤ a ≤ 11\)
\(4 ≤ b < 8\)
(a)Work out the largest possible value of \(-3a + 5b\).
(b)\(x\) and \(y\) are both negative numbers.
Show that \(x ÷ y\) could equal \(12\).
Show answer
(a) \(29\)
(b) \(x = -36\), \(y = -3\) gives \(-36 ÷ (-3) = 12\)A strict inequality does not reach its boundary, so the whole numbers allowed are \(a = 2\) to \(11\) and \(b = 4\) to \(7\).
The coefficient of \(a\) is \(-3\), so use the smallest value \(a = 2\): \(-3 \times 2 = -6\).
The coefficient of \(b\) is \(5\), so use the largest value \(b = 7\): \(5 \times 7 = 35\).
Largest value: \(-6 + 35 = 29\)
Both numbers must be negative, so try \(x = -36\) and \(y = -3\).
A negative number divided by a negative number is positive:
\(-36 ÷ (-3) = 12\)
So \(x ÷ y\) could equal \(12\).
- (a) M1the whole-number values \(a = 2\) and \(b = 7\) that make the expression largest
- (a) A1largest possible value \(29\)
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Either paper 3 marksTick one box for each statement.
True May be true Not true (i) If a whole number is positive, the number is \(> 1\). ☐ ☐ ☐ (ii) If a number (not necessarily whole) is \(> -1\), the number is \(≥ 0\). ☐ ☐ ☐ (iii) If a number (not necessarily whole) is \(> 8\), the number is \(10\). ☐ ☐ ☐ Show answer
Ticks: (i) may be true, (ii) may be true, (iii) may be trueFor each statement, find values that satisfy the condition and test the claim on them.
(i) Both \(1\) and \(2\) are positive. The claim is false for \(1\) and true for \(2\). One allowed value satisfies the claim and another does not, so the statement may be true.
(ii) Both \(-0.5\) and \(0\) satisfy \(> -1\). The claim \(≥ 0\) is false for \(-0.5\) and true for \(0\). One allowed value satisfies the claim and another does not, so the statement may be true.
(iii) The boundary value \(8\) is not included, because the sign is strict. The claimed value is \(10\). This value satisfies the condition, but the number could also take other values, so the statement may be true.
Answer: Ticks: (i) may be true, (ii) may be true, (iii) may be true
- B1(i) may be true: testing 1 and 2
- B1(ii) may be true: testing -0.5 and 0
- B1(iii) may be true: comparing the claimed value 10 with the boundary 8
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Either paper 1 mark
\(x\) is greater than or equal to \(-5\) and smaller than \(4\).
Circle the inequality that shows this.
A: \(-5 ≤ x ≤ 4\)
B: \(-5 ≤ x < 4\)
C: \(-5 < x < 4\)
D: \(-5 < x ≤ 4\)
Show answer
B: \(-5 ≤ x < 4\)"greater than or equal to \(-5\)" gives the lower end: the sign next to \(-5\) is \(≤\), and \(<\) leaves \(-5\) out while \(≤\) includes it.
"smaller than \(4\)" gives the upper end: the sign next to \(4\) is \(<\).
Write the smaller number, \(-5\), on the left and the larger number, \(4\), on the right:
\(-5 ≤ x < 4\)
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Either paper 1 mark
Circle the list of all the integers that satisfy
\(-3 < n ≤ 3\)
A: \(-3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3\)
B: \(-2,\ -1,\ 0,\ 1,\ 2\)
C: \(-3,\ -2,\ -1,\ 0,\ 1,\ 2\)
D: \(-2,\ -1,\ 0,\ 1,\ 2,\ 3\)
Show answer
\(-2,\ -1,\ 0,\ 1,\ 2,\ 3\)A sign ≤ keeps the end value, and a sign \(<\) drops it.
Lower end: \(<\) so -3 is not included. Upper end: \(≤\) so 3 is included.
The integers run from -2 to 3, with every whole number in between:
\(-2,\ -1,\ 0,\ 1,\ 2,\ 3\)