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42 questions at your level
Difficulty
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Level 1
Level 2
Level 3
Level 4
Level 5
Linear inequalities
16 questions
Lesson
Not started
Mark as done
Write down all the integer values of
x
x
x
that satisfy
−
2
<
x
≤
2
-2 < x \le 2
−
2
<
x
≤
2
●
●●●●
Level 1
1 mark
Start
→
Mark as done
Solve
2
x
+
3
>
9
2x + 3 > 9
2
x
+
3
>
9
●
●●●●
Level 1
1 mark
Start
→
Mark as done
Solve the inequality
3
x
+
5
≤
20
3x + 5 \le 20
3
x
+
5
≤
20
●●
●●●
Level 2
2 marks
Start
→
Mark as done
Solve
x
2
+
5
>
9
\frac{x}{2} + 5 > 9
2
x
+
5
>
9
(2 marks)
●●
●●●
Level 2
2 marks
Start
→
Mark as done
(a)
Solve
3
x
+
5
>
2
x
−
1
3x + 5 > 2x - 1
3
x
+
5
>
2
x
−
1
[2]
(b)
Write down the smallest integer that satisfies the inequality. [1]
●●
●●●
Level 2
3 marks
Start
→
Mark as done
Solve the inequality
4
x
−
3
<
17
4x - 3 < 17
4
x
−
3
<
17
●●●
●●
Level 3
2 marks
Start
→
Mark as done
n
n
n
is an integer and
−
2
<
n
≤
3
-2 < n \le 3
−
2
<
n
≤
3
Write down all the possible values of
n
n
n
.
●●●
●●
Level 3
2 marks
Start
→
Mark as done
n
n
n
is an integer.
n
n
n
satisfies
both
of the inequalities
−
4
<
n
≤
1
and
n
>
−
2
-4 < n \le 1 \qquad \text{and} \qquad n > -2
−
4
<
n
≤
1
and
n
>
−
2
Write down all the possible values of
n
n
n
.
●●●
●●
Level 3
2 marks
Start
→
Mark as done
(a)
Solve
−
4
≤
2
x
+
2
<
10
-4 \le 2x + 2 < 10
−
4
≤
2
x
+
2
<
10
(3 marks)
(b)
Write down all the integer values of
x
x
x
that satisfy the inequality in part (a). (1 mark)
●●●
●●
Level 3
4 marks
Start
→
Mark as done
−
4
<
n
≤
3
-4 < n \le 3
−
4
<
n
≤
3
n
n
n
is an integer.
(a)
Write down the least possible value of
n
n
n
.
(1 mark)
(b)
On the number line below, show the inequality
−
2
≤
y
<
4
-2 \le y < 4
−
2
≤
y
<
4
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
y
(2 marks)
(c)
Solve
3
4
w
+
5
<
17
\dfrac{3}{4}w + 5 < 17
4
3
w
+
5
<
17
(3 marks)
●●●
●●
Level 3
6 marks
Start
→
Mark as done
(a)
Solve the inequality
7
−
2
x
>
1
7 - 2x > 1
7
−
2
x
>
1
(b)
Write down the largest integer that satisfies
7
−
2
x
>
1
7 - 2x > 1
7
−
2
x
>
1
●●●●
●
Level 4
3 marks
Start
→
Mark as done
(a)
Solve the inequality
3
<
2
x
+
1
≤
9
3 < 2x + 1 \le 9
3
<
2
x
+
1
≤
9
(b)
x
x
x
is an integer. Write down all the values of
x
x
x
that satisfy
3
<
2
x
+
1
≤
9
3 < 2x + 1 \le 9
3
<
2
x
+
1
≤
9
●●●●
●
Level 4
3 marks
Start
→
Mark as done
A van-hire company charges a fixed fee of £40 plus £22 per day.
Amy can spend
at most
£150.
Work out the greatest number of whole days Amy can hire the van for.
●●●●
●
Level 4
3 marks
Start
→
Mark as done
Solve the inequality
5
−
3
x
≤
−
7
5 - 3x \le -7
5
−
3
x
≤
−
7
●●●●
●
Level 4
2 marks
Start
→
Mark as done
(a)
Solve the inequality
−
5
≤
3
x
+
1
<
10
-5 \le 3x + 1 < 10
−
5
≤
3
x
+
1
<
10
(b)
x
x
x
is an integer. Write down all the values of
x
x
x
that satisfy
−
5
≤
3
x
+
1
<
10
-5 \le 3x + 1 < 10
−
5
≤
3
x
+
1
<
10
●●●●
●
Level 4
3 marks
Start
→
Mark as done
A rectangle has width
w
w
w
cm and length
(
w
+
7
)
(w + 7)
(
w
+
7
)
cm.
The perimeter of the rectangle must be
less than
46
46
46
cm.
(a)
Show that
w
<
8
w < 8
w
<
8
(b)
The width is a whole number of centimetres. Write down the largest possible width.
●●●●●
Level 5
4 marks
Start
→
Quadratic inequalities
15 questions
Lesson
Not started
Mark as done
Solve the inequality
x
2
≤
16
x^2 \le 16
x
2
≤
16
●●●●
●
Level 4
2 marks
Start
→
Mark as done
n
n
n
is an integer and
n
2
<
30
n^2 < 30
n
2
<
30
(a)
Write down the greatest possible value of
n
n
n
.
(b)
Write down the least possible value of
n
n
n
.
●●●●
●
Level 4
3 marks
Start
→
Mark as done
Solve the inequality
x
2
<
49
x^2 < 49
x
2
<
49
●●●●
●
Level 4
2 marks
Start
→
Mark as done
Solve
x
2
−
3
x
−
4
≤
0
x^2 - 3x - 4 \le 0
x
2
−
3
x
−
4
≤
0
(4 marks)
●●●●
●
Level 4
4 marks
Start
→
Mark as done
Solve
x
2
−
5
x
−
14
>
0
x^2 - 5x - 14 > 0
x
2
−
5
x
−
14
>
0
(4 marks)
●●●●
●
Level 4
4 marks
Start
→
Mark as done
Solve
(
2
x
+
7
)
(
4
x
−
3
)
<
0
(2x + 7)(4x - 3) < 0
(
2
x
+
7
)
(
4
x
−
3
)
<
0
(2 marks)
●●●●
●
Level 4
2 marks
Start
→
Mark as done
Solve the inequality
x
2
−
3
x
−
10
>
0
x^2 - 3x - 10 > 0
x
2
−
3
x
−
10
>
0
●●●●●
Level 5
3 marks
Start
→
Mark as done
Solve the inequality
x
2
+
x
−
12
<
0
x^2 + x - 12 < 0
x
2
+
x
−
12
<
0
●●●●●
Level 5
3 marks
Start
→
Mark as done
Solve the inequality
x
2
>
3
x
x^2 > 3x
x
2
>
3
x
●●●●●
Level 5
3 marks
Start
→
Mark as done
Solve the inequality
x
2
−
5
x
+
6
≥
0
x^2 - 5x + 6 \ge 0
x
2
−
5
x
+
6
≥
0
●●●●●
Level 5
3 marks
Start
→
Mark as done
Solve the inequality
x
2
+
4
x
−
21
≤
0
x^2 + 4x - 21 \le 0
x
2
+
4
x
−
21
≤
0
●●●●●
Level 5
3 marks
Start
→
Mark as done
(a)
Solve the inequality
x
2
−
2
x
−
8
>
0
x^2 - 2x - 8 > 0
x
2
−
2
x
−
8
>
0
(b)
Write down the smallest
positive integer
that satisfies
x
2
−
2
x
−
8
>
0
x^2 - 2x - 8 > 0
x
2
−
2
x
−
8
>
0
●●●●●
Level 5
4 marks
Start
→
Mark as done
n
n
n
is an integer that satisfies
both
of the inequalities
n
2
<
40
and
2
n
+
3
>
8
n^2 < 40 \qquad \text{and} \qquad 2n + 3 > 8
n
2
<
40
and
2
n
+
3
>
8
Write down all the possible values of
n
n
n
.
●●●●●
Level 5
3 marks
Start
→
Mark as done
Find the set of possible values of
x
x
x
for which
9
x
2
−
49
<
0
9x^2 - 49 < 0
9
x
2
−
49
<
0
\qquad and \qquad
15
−
7
x
−
2
x
2
>
0
15 - 7x - 2x^2 > 0
15
−
7
x
−
2
x
2
>
0
You must show all your working.
(5 marks)
●●●●●
Level 5
5 marks
Start
→
Mark as done
Find the set of possible values of
x
x
x
for which
x
2
+
2
x
>
24
x^2 + 2x > 24
x
2
+
2
x
>
24
\qquad and \qquad
2
x
2
<
7
x
+
30
2x^2 < 7x + 30
2
x
2
<
7
x
+
30
You must show all your working.
(5 marks)
●●●●●
Level 5
5 marks
Start
→
Inequalities as regions on a graph
11 questions
Lesson
Not started
Mark as done
The diagram shows the line
x
=
2
x = 2
x
=
2
and the line
y
=
1
y = 1
y
=
1
.
y = 1
x = 2
R
x
y
Diagram NOT accurately drawn
The region
R
R
R
is to the right of the line
x
=
2
x = 2
x
=
2
and below the line
y
=
1
y = 1
y
=
1
.
Points on the line
x
=
2
x = 2
x
=
2
are included in
R
R
R
; points on the line
y
=
1
y = 1
y
=
1
are not.
Write down the two inequalities that define
R
R
R
.
●●●
●●
Level 3
2 marks
Start
→
Mark as done
A region is defined by the three inequalities
x
≥
0
,
y
<
x
+
1
,
x
+
y
≤
6
x \ge 0, \qquad y < x + 1, \qquad x + y \le 6
x
≥
0
,
y
<
x
+
1
,
x
+
y
≤
6
Here are three points:
A
(
2
,
1
)
B
(
4
,
3
)
C
(
1
,
4
)
\;A(2, 1) \qquad B(4, 3) \qquad C(1, 4)
A
(
2
,
1
)
B
(
4
,
3
)
C
(
1
,
4
)
Which of the points lies in the region? You must show how you decide.
●●●
●●
Level 3
3 marks
Start
→
Mark as done
A region
R
R
R
is defined by the two inequalities
x
<
3
and
y
≥
2
x < 3 \qquad \text{and} \qquad y \ge 2
x
<
3
and
y
≥
2
x
y
(a)
Is the point
(
1
,
4
)
(1, 4)
(
1
,
4
)
in the region
R
R
R
? Give a reason.
(b)
Is the point
(
3
,
2
)
(3, 2)
(
3
,
2
)
in the region
R
R
R
? Give a reason.
●●●
●●
Level 3
3 marks
Start
→
Mark as done
The diagram shows three straight lines and a region
R
R
R
bounded by them.
-1
1
2
3
4
5
6
-1
1
2
3
4
5
y = x + 1
x + y = 5
y = 1
R
x
y
Diagram NOT accurately drawn
All three boundary lines are included in
R
R
R
.
Write down the three inequalities that define
R
R
R
.
●●●●
●
Level 4
3 marks
Start
→
Mark as done
The diagram shows the lines
y
=
x
y = x
y
=
x
,
y
=
3
y = 3
y
=
3
and
x
=
2
x = 2
x
=
2
, and a region
R
R
R
bounded by them.
y = x
y = 3
x = 2
R
x
y
Diagram NOT accurately drawn
All three boundary lines are included in
R
R
R
.
Write down the three inequalities that define
R
R
R
.
●●●●
●
Level 4
3 marks
Start
→
Mark as done
A triangular region is bounded by the three inequalities
y
≥
0
,
y
≤
2
x
,
x
≤
3
y \ge 0, \qquad y \le 2x, \qquad x \le 3
y
≥
0
,
y
≤
2
x
,
x
≤
3
x
y
Work out the coordinates of the three vertices of the region.
●●●●
●
Level 4
3 marks
Start
→
Mark as done
The diagram shows the vertical line
x
=
−
1
x = -1
x
=
−
1
(solid), the line
y
=
x
−
2
y = x - 2
y
=
x
−
2
(dashed), and the horizontal line
y
=
4
y = 4
y
=
4
(solid). The region
R
R
R
lies to the right of the vertical line, above the dashed line, and below the horizontal line.
R
x
y
Write down the three inequalities that define the region
R
R
R
.
(A dashed line is not included in the region; a solid line is.)
●●●●
●
Level 4
3 marks
Start
→
Mark as done
A region is defined by the three inequalities
x
≥
0
,
y
≥
0
,
x
+
y
<
4
x \ge 0, \qquad y \ge 0, \qquad x + y < 4
x
≥
0
,
y
≥
0
,
x
+
y
<
4
How many points with
integer coordinates
lie in the region?
●●●●
●
Level 4
3 marks
Start
→
Mark as done
(a)
On the grid, show by shading, the region that satisfies all of these inequalities.
y
>
−
1
x
<
2
y
<
2
x
+
1
x
+
y
>
−
3
y > -1 \qquad x < 2 \qquad y < 2x + 1 \qquad x + y > -3
y
>
−
1
x
<
2
y
<
2
x
+
1
x
+
y
>
−
3
Label the region
R
R
R
.
-4
-3
-2
-1
1
2
3
4
5
-4
-3
-2
-1
1
2
3
4
5
6
x
y
(4 marks)
Aisha says,
"One of these four inequalities is not needed.
R
R
R
would be the same without it."
Aisha is correct.
(b)
Which inequality is not needed?
(1 mark)
●●●●
●
Level 4
5 marks
Start
→
Mark as done
A region is defined by the three inequalities
x
≥
1
,
y
≤
x
+
1
,
x
+
y
<
8
x \ge 1, \qquad y \le x + 1, \qquad x + y < 8
x
≥
1
,
y
≤
x
+
1
,
x
+
y
<
8
The point
(
a
,
3
)
(a, 3)
(
a
,
3
)
lies in the region, where
a
a
a
is an integer.
Find all the possible values of
a
a
a
.
●●●●●
Level 5
3 marks
Start
→
Mark as done
A region is defined by the three inequalities
y
≥
1
,
y
<
2
x
+
1
,
x
+
y
≤
7
y \ge 1, \qquad y < 2x + 1, \qquad x + y \le 7
y
≥
1
,
y
<
2
x
+
1
,
x
+
y
≤
7
The point
(
2
,
b
)
(2, b)
(
2
,
b
)
, where
b
b
b
is an
integer
, lies in the region.
Write down all the possible values of
b
b
b
.
●●●●●
Level 5
3 marks
Start
→