think! Mathematics Textbook Secondary 3 (8th Edition) solutions
Ex 2A
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Solutions
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(a)
\begin{align} 3x & < x - 7 \\ 3x - x & < - 7 \\ 2x & < - 7 \\ x & < {-7 \over 2} \\ x & < -3.5 \end{align}
(b)
\begin{align} 2x + 8 & > 9x \\ 2x - 9x & > - 8 \\ -7x & > - 8 \\ x & < {-8 \over -7} \\ x & < {8 \over 7} \end{align}
(c)
\begin{align} 5x & \ge 3(x - 4) \\ 5x & \ge 3x - 12 \\ 5x - 3x & \ge -12 \\ 2x & \ge -12 \\ x & \ge {-12 \over 2} \\ x & \ge -6 \end{align}
(d)
\begin{align} 4x - 2 & \le 7x - 5 \\ 4x - 7x & \le -5 + 2 \\ -3x & \le -3 \\ x & \ge {-3 \over -3} \\ x & \ge 1 \end{align}
(a)
\begin{align}
x - 4 & \le 3
&&&
3x & \ge -6 \\
x & \le 3 + 4
&&&
x & \ge {-6 \over 3} \\
x & \le 7
&&&
x & \ge - 2
\end{align}
$$ -2 \le x \le 7 $$
(b)
\begin{align}
2x + 5 & < 15
&&&
3x - 2 & > -6 \\
2x & < 15 - 5
&&&
3x & > -6 + 2 \\
2x & < 10
&&&
3x & > -4 \\
x & < {10 \over 2}
&&&
x & > -{4 \over 3} \\
x & < 5
\end{align}
$$ -{4 \over 3} < x < 5 $$
(a)
\begin{align}
5x - 1 & < 4
&&&
3x + 5 & \ge x + 1 \\
5x & < 4 + 1
&&&
3x - x & \ge 1 - 5 \\
5x & < 5
&&&
2x & \ge - 4 \\
x & < {5 \over 5}
&&&
x & \ge {-4 \over 2} \\
x & < 1
&&&
x & \ge -2
\end{align}
\begin{align}
-2 & \phantom{.} \le x < 1 \\
\\
\text{Integer values of } & x \text{ are } -2, -1, 0
\end{align}
(b)
\begin{align}
2x - 5 & \ge 1
&&&
3x - 1 & < 26 \\
2x & \ge 1 + 5
&&&
3x & < 26 + 1 \\
2x & \ge 6
&&&
3x & < 27 \\
x & \ge {6 \over 2}
&&&
x & < {27 \over 3} \\
x & \ge 3
&&&
x & < 9
\end{align}
\begin{align}
3 & \phantom{.} \le x < 9 \\
\\
\text{Integer values of } & x \text{ are } 3, 4, 5, 6, 7, 8
\end{align}
(a)
\begin{align} -4 & \le 2x &&& 2x & \le 3x - 2 \\ {-4 \over 2} & \le x &&& 2x - 3x & \le -2 \\ -2 & \le x &&& -x & \le -2 \\ & &&& x & \ge 2 \end{align}
$$ x \ge 2 $$
(b)
\begin{align} 1 - x & < -2 &&& -2 & \le 3 - x \\ - x & < -2 - 1 &&& x & \le 3 + 2 \\ -x & < - 3 &&& x & \le 5 \\ x & > 3 \end{align}
$$ 3 < x \le 5 $$
(c)
\begin{align} 3x - 3 & < x - 9 &&& x - 9 & < 2x \\ 3x - x & < -9 + 3 &&& x - 2x & < 9 \\ 2x & < -6 &&& -x & < 9 \\ x & < {-6 \over 2} &&& x & > - 9 \\ x & < -3 \end{align}
$$ -9 < x < -3 $$
\begin{align}
\text{Let } x \text{ represent } & \text{the number of apples} \\
\\
55 \text{ cents} & = \$ 0.55 \\
\\
\text{Total revenue} & = 0.55 \times x \\
& = 0.55x \\
\\
0 < \text{Profit} & \le \$ 20
\phantom{000000} [\text{'did not make a profit of more than \$ 20'}] \\ \\
0 < 0.55x & - 66.5 \le 20
\end{align}
\begin{align}
0 & < 0.55x - 66.5
&&&
0.55x - 66.5 & \le 20 \\
66.5 & < 0.55x
&&&
0.55x & \le 20 + 66.5 \\
{66.5 \over 0.55} & < x
&&&
0.55x & \le 86.5 \\
120{10 \over 11} & < x
&&&
x & \le {86.5 \over 0.55} \\
& &&&
x & \le 157{3 \over 11}
\end{align}
\begin{align}
120{10 \over 11} & < x \le 157{3 \over 11} \\
\\
\therefore \text{He could have sold } &
\text{between } 121 \text{ and } 157 \text{ apples (inclusive)}
\end{align}
\begin{align}
{1 \over 2}x - 4 & > {1 \over 3}x
&&&
{1 \over 6}x + 1 & < {1 \over 8}x + 3
\\
{1 \over 2}x - {1 \over 3}x & > 4
&&&
{1 \over 6}x - {1 \over 8}x & < 3 - 1
\\
{1 \over 6}x & > 4
&&&
{1 \over 24}x & < 2
\\
x & > 6(4)
&&&
x & < 24(2)
\\
x & > 24
&&&
x & < 48
\end{align}
\begin{align}
24 & \phantom{.} < x < 48 \\
\\
\text{Prime numbers: } & 29, 31, 37, 41, 43, 47
\end{align}
\begin{align}
x + 2 & < 5 \sqrt{17}
&&&
5 \sqrt{17} & < x + 3 \\
x & < 5 \sqrt{17} - 2
&&&
5 \sqrt{17} - 3 & < x \\
x & < 18.615
&&&
17.615 & < x
\end{align}
\begin{align}
17.615 & \phantom{.} < x < 18.615 \\
\\
\text{Integer value} & \text{ of } x = 18
\end{align}
(a)
\begin{align}
3 - a & \le a - 4
&&&
a - 4 & \le 9 - 2a \\
3 + 4 & \le a + a
&&&
a + 2a & \le 9 + 4 \\
7 & \le 2a
&&&
3a & \le 13 \\
{7 \over 2} & \le a
&&&
a & \le {13 \over 3}
\end{align}
$$ {7 \over 2} \le a \le {13 \over 3} $$
(b)
\begin{align}
1 - b & < b - 1
&&&
b - 1 & < 11 - 2b \\
1 + 1 & < b + b
&&&
b + 2b & < 11 + 1 \\
2 & < 2b
&&&
3b & < 12 \\
{2 \over 2} & < b
&&&
b & < {12 \over 3} \\
1 & < b
&&&
b & < 4
\end{align}
$$ 1 < b < 4 $$
(c)
\begin{align}
3 - c & < 2c - 1
&&&
2c - 1 & < 5 + c \\
3 + 1 & < 2c + c
&&&
2c - c & < 5 + 1 \\
4 & < 3c
&&&
c & < 6 \\
{4 \over 3} & < c
\end{align}
$$ {4 \over 3} < c < 6 $$
(d)
\begin{align}
3d - 5 & < d + 1
&&&
d + 1 & \le 2d + 1 \\
3d - d & < 1 + 5
&&&
d - 2d & \le 1 - 1 \\
2d & < 6
&&&
-d & \le 0 \\
d & < {6 \over 2}
&&&
d & \ge 0 \\
d & < 3
\end{align}
$$ 0 \le d < 3 $$
(a)
\begin{align}
{a \over 4} + 3 & \le 4
&&&
4 & \le {a \over 4} + 6 \\
{a \over 4} & \le 4 - 3
&&&
4 - 6 & \le {a \over 4} \\
{a \over 4} & \le 1
&&&
-2 & \le {a \over 4} \\
{a \over 4} & \le {4 \over 4}
&&&
{-8 \over 4} & \le {a \over 4} \\
a & \le 4
&&&
-8 & \le a
\end{align}
$$ -8 \le a \le 4 $$
(b)
\begin{align} {b \over 3} & \ge {b \over 2} + 1 &&& {b \over 2} + 1 & \ge b - 1 \\ {b \over 3} - {b \over 2} & \ge 1 &&& {b \over 2} - b & \ge - 1 - 1 \\ {2b \over 6} - {3b \over 6} & ge {6 \over 6} &&& {b \over 2} - {2b \over 2} & \ge - 2 \\ {-b \over 6} & \ge {6 \over 6} &&& {-b \over 2} & \ge {-4 \over 2} \\ -b & \ge 6 &&& -b & \ge -4 \\ b & \le -6 &&& b & \le 4 \end{align}
$$ b \le - 6 $$
(c)
\begin{align}
2(1 - c) & > c - 1
&&&
c - 1 & \ge {c - 2 \over 7} \\
2 - 2c & > c - 1
&&&
{7(c - 1) \over 7} & \ge {c -2 \over 7} \\
2 + 1 & > c + 2c
&&&
7(c - 1) & \ge c - 2 \\
3 & > 3c
&&&
7c - 7 & \ge c - 2 \\
{3 \over 3} & > c
&&&
7c - c & \ge - 2 + 7 \\
1 & > c
&&&
6c & \ge 5 \\
1 & > c
&&&
c & \ge {5 \over 6}
\end{align}
$$ {5 \over 6} \le c < 1 $$
(d)
\begin{align}
d - 5 & < {2d \over 5}
&&&
{2d \over 5} & \le {d \over 2} + {1 \over 5} \\
{5(d - 5) \over 5} & < {2d \over 5}
&&&
{4d \over 10} & \le {5d \over 10} + {2 \over 10} \\
5(d - 5) & < 2d
&&&
{4d \over 10} & \le {5d + 2 \over 10} \\
5d - 25 & < 2d
&&&
4d & \le 5d + 2 \\
5d - 2d & < 25
&&&
4d - 5d & \le 2 \\
3d & < 25
&&&
-d & \le 2 \\
d & < {25 \over 3}
&&&
d & \ge -2
\end{align}
$$ -2 \le d < {25 \over 3} $$
To solve this question, we have to assume the present costs $210 (the most expensive possibility)
\begin{align}
\text{Let } \$ x \text{ represent } & \text{the money Weiming pays} \\
\\
\text{Amount Joyce pays} & = \$ (210 - x)
\end{align}
\begin{align}
\text{Weiming's share} & > 2 \times \text{Joyce's share}
&&& \text{Weiming's share} - \text{Joyce's share} & \le \$ 150 \\
x & > 2(210 - x)
&&&
x - (210 - x) & \le 150 \\
x & > 420 - 2x
&&&
x - 210 + x & \le 150 \\
x + 2x & > 420
&&&
2x & \le 150 + 210 \\
3x & > 420
&&&
2x & \le 360 \\
x & > {420 \over 3}
&&&
x & \le {360 \over 2} \\
x & > 140
&&&
x & \le 180
\end{align}
\begin{align}
140 & \phantom{.} < x \le 180 \\
\\
\text{Greatest amount} & \text{ Weiming pays} = \$180
\end{align}
(a) Both inequalities must satisfy x < - 2
\begin{align}
ax + 3 & > 2x + b
&&&
cx - 1 & > 5x - 4 \\
ax - 2x & > b - 3
&&&
cx - 5x & > - 4 + 1 \\
(a - 2)x & > b - 3
&&&
(c - 5)x & > - 3 \\
\\
\text{If } a = 1,& \phantom{.} b =5,
&&&
\text{If } & c = 4, \\
(1 - 2)x & > 5 - 3
&&&
(4 - 5)x & > - 3 \\
- x & > 2
&&&
-x & > - 3 \\
x & < - 2
&&&
x & < 3 \phantom{000000} [\text{Both satisfies } x < -2]
\end{align}
$$ \therefore a = 1, b = 5, c = 4 $$
(b) One equality must satisfy x > -3 and the other equality must satisfy x < 4
\begin{align}
(a - 2)x & > b - 3
&&&
(c - 5)x & > - 3 \\
\\
\text{If } a = 1 &, b = -1,
&&&
\text{If } c & = 6, \\
(1 - 2)x & > -1 - 3
&&&
(6 - 5)x & > -3 \\
-x & > - 4
&&&
x & > - 3 \\
x & < 4
\end{align}
$$ \therefore a = 1, b = -1, c = 6 $$
(c)
\begin{align}
(a - 2)x & > b - 3
&&&
(c - 5)x & > - 3 \\
\\
\text{If } a = 3 &, b = 6,
&&&
\text{If } c & = 4, \\
(3 - 2)x & > 6 - 3
&&&
(4 - 5)x & > -3 \\
x & > 3
&&&
-x & > - 3 \\
&
&&&
x & < 3
\end{align}
$$ \therefore a = 3, b = 6, c = 4 $$
\begin{align} \text{Let } x \text{ represent } & \text{no. of bubble tea} \\ \\ \text{Total cost of bubble tea} & = 3.20 \times x \\ & = \$ 3.2x \\ \\ \text{No. of ice-cream cone} & = 12 - x \\ \\ \text{Total cost of ice-cream cone} & = 2.4 \times (12 - x) \\ & = 2.4(12 - x) \\ & = \$ (28.8 - 2.4x) \\ \\ 3.2 x + 28.8 - 2.4x & \le 30 \\ 0.8x & \le 30 - 28.8 \\ 0.8x & \le 1.2 \\ x & \le {1.2 \over 0.8} \\ x & \le 1.5 \\ \\ \text{Only possible to buy } & \text{0 or 1 cup of bubble tea} \\ \\ \therefore \text{She is not able to buy more} & \text{ bubble tea than ice-cream cones} \end{align}