[4 marks sum]

Question 14 Marks

If $P = \{x : 7x - 4 > 5x + 2, x \in R\}$ and $Q - \{x : x - 19 \geq 1 - 3x, x \in R\},$ represent the following solution set on different number lines:
$P' \cap Q$

Answer

$P=\{x: 7 x-4>5 x+2, x \in R\}$
$7 x-4>5 x+2$
$7 x-5 x>2+4$
$2 x>6$
$X>3$
$P=\{4,5,6,7, \ldots \ldots . .\}$
$\text { and }$
$Q-\{x: x-19 \geq 1-3 x, x \in R\}$
$x-19 \geq 1-3 x$
$x+3 x \geq 1+19$
$P x \geq 20$
$x \geq s$
$Q=\{5,6,7,8,9 \ldots \ldots . . .\}$
$P=\{\phi\}$

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Question 24 Marks

If P = {x : 7x - 4 > 5x + 2, x ∈ R} and Q - {x : x - 19 ≥ 1 - 3x, x ∈ R}, represent the following solution set on different number lines:
P ∩ Q

Answer

$
\begin{aligned}
& P=\{x: 7 x-4>5 x+2, x \in R\} \\
& 7 x-4>5 x+2 \\
& 7 x-5 x>2+4 \\
& 2 x>6 \\
& x>3
\end{aligned}
$
$
p=\{4,5,6,7, \ldots \ldots \ldots\}
$
and
$
\begin{aligned}
& Q-\{x: x-19 \geq 1-3 x, x \in R\} \\
& x-19 \geq 1-3 x \\
& x+3 x \geq 1+19 \\
& 4 x \geq 20 \\
& x \geq 5
\end{aligned}
$
$
\begin{aligned}
& Q=\{5,6,7,8,9 \ldots \ldots .\} \\
& P \cap Q=\{5,6,7,8, \ldots \ldots . . .\}
\end{aligned}
$

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Question 34 Marks

If P = {x : 7x - 2 > 4x + 1, x ∈ R} and Q = {x : 9x - 45 ≥ 5 (x -5),x ∈ R} , represent the following solution set on different number lines:
P ∩ Q'

Answer

$
\begin{aligned}
& P=\{x: 7 x-2>4 x+1, x \in R\} \\
& 7 x-2>4 x+1 \\
& 7 X-4 X>1+2 \\
& 3 x>3 \\
& X>1
\end{aligned}
$
$
p=\{2,3,4,5, \ldots \ldots \ldots . .\}
$
and
$
\begin{aligned}
& Q=\{x: 9 x-45 \geq 5(x-5), x \in R\} \\
& 9 x-45 \geq 5 x-25 \\
& 9 x-5 x \geq-25+45 \\
& 4 x \geq 20 \\
& x \geq 5
\end{aligned}
$
$
\begin{aligned}
& Q=\{5,6,7,8,9, \ldots \ldots \ldots . . .\} \\
& P \cap Q^{\prime}=\{6,7,8,9, \ldots \ldots \ldots . . .\}
\end{aligned}
$

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Question 44 Marks

If P = {x : 7x - 2 > 4x + 1, x ∈ R} and Q = {x : 9x - 45 ≥ 5 (x -5),x ∈ R} , represent the following solution set on different number lines:
P - Q

Answer

$
\begin{aligned}
& P=\{x: 7 x-2>4 x+1, x \in R\} \\
& 7 x-2>4 x+1 \\
& 7 x-4 x>1+2 \\
& 3 x>3 \\
& x>1
\end{aligned}
$
$
p=\{2,3,4,5, \cdots \cdots \cdots \cdots\}
$
and
$
\begin{aligned}
& Q=\{x: 9 x-45 \geq 5(x-5), x \in R\} \\
& 9 x-45 \geq 5 x-25 \\
& 9 x-5 x \geq-25+45 \\
& 4 x \geq 20 \\
& x \geq 5
\end{aligned}
$
$
\begin{aligned}
& Q=\{5,6,7,8,9, \ldots \ldots \ldots . . .\} \\
& P-Q=\{6,7,8,9 \ldots \ldots \ldots . \}
\end{aligned}
$

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Question 54 Marks

If P = {x : 7x - 2 > 4x + 1, x ∈ R} and Q = {x : 9x - 45 ≥ 5 (x -5),x ∈ R} , represent the following solution set on different number lines:
P ∩ Q

Answer

$
\begin{aligned}
& P=\{x: 7 x-2>4 x+1, x \in R\} \\
& 7 x-2>4 x+1 \\
& 7 X-4 X>1+2 \\
& 3 x>3 \\
& x>1
\end{aligned}
$
$
p=\{2,3,4,5, \cdots \cdots \cdots \cdots\}
$
and
$
\begin{aligned}
& Q=\{x: 9 x-45 \geq 5(x-5), x \in R\} \\
& 9 x-45 \geq 5 x-25 \\
& 9 x-5 x \geq-25+45 \\
& 4 x \geq 20 \\
& x \geq 5
\end{aligned}
$
$
\begin{aligned}
& Q=\{5,6,7,8,9, \ldots \ldots \ldots . . .\} \\
& P \cap Q=\{2,3,4,5\}
\end{aligned}
$

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Question 64 Marks

If P = { x : -3 < x ≤ 7, x ∈ R} and Q = { x : - 7 ≤ x < 3, x ∈ R} , represent the following solution set on the different number lines:
P-Q

Answer

$
\begin{aligned}
& P=\{x:-3<x \leq 7, x \in R\} \text { and } Q=\{x:-7 \leq x<3, x \in R\} \\
& P=\{-2,-1,0,1,2,3,4,5,6,7\} \text { and } Q=\{-7,-6,-5,-4,-3,-2,-1,0,1,2\} \\
& P-Q=\{3,4,5,6,7\}
\end{aligned}
$

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Question 74 Marks

If P = { x : -3 < x ≤ 7, x ∈ R} and Q = { x : - 7 ≤ x < 3, x ∈ R} , represent the following solution set on the different number lines:
Q' ∩ P

Answer

$
\begin{aligned}
& P=\{x:-3<x \leq 7, x \in R\} \text { and } Q=\{x:-7 \leq x<3, x \in R\} \\
& P=\{-2,-1,0,1,2,3,4,5,6,7\} \text { and } Q=\{-7,-6,-5,-4,-3,-2,-1,0,1,2\} \\
& Q^{\prime} \cap P=\{3,4,5,6,7\}
\end{aligned}
$

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Question 84 Marks

If P = { x : -3 < x ≤ 7, x ∈ R} and Q = { x : - 7 ≤ x < 3, x ∈ R} , represent the following solution set on the different number lines :
P ∩ Q

Answer

$
\begin{aligned}
& P=\{x:-3<x \leq 7, x \in R\} \text { and } Q=\{x:-7 \leq x<3, x \in R\} \\
& P=\{-2,-1,0,1,2,3,4,5,6,7\} \text { and } Q=\{-7,-6,-5,-4,-3,-2,-1,0,1,2\} \\
& \text { (i) } P \cap Q \\
& P \cap Q=\{-2,-1,0,1,2\}
\end{aligned}
$

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Question 94 Marks

Solve the following linear in$-$equation and graph the solution set on a real number line: $\frac{5}{4} x>1+\frac{1}{3}(4 x-1), x \in R$

Answer

$\frac{5}{4} x>1+\frac{1}{3}(4 x-1)$
$\frac{5}{4} x>\frac{3+(4 x-1)}{3}$
$15 x>12+16 x-4$
$15 x-16 x>8$
$-x>8$
$x<-8$
Solution set $=[x<-8]$

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Question 104 Marks

Solve the following linear in$-$equation and graph the solution set on a real number line: $\frac{1}{3}(5 x-8) \geq \frac{1}{2}(4 x-7), x \in R$

Answer

$\frac{1}{3}(5 x-8) \geq \frac{1}{2}(4 x-7)$
$2(5 x-8) \geq 3(4 x-7)$
$10 x-16 \geq 12 x-21$
$10 x-12 x \geq-21+16$
$-2 x \geq-5$
$x \leq \frac{5}{2}$
$x \leq 2 \frac{1}{2}$
$\text { Solution set }=\left[x \leq 2 \frac{1}{2}\right]$

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Question 114 Marks

Solve the following linear in$-$equation and graph the solution set on a real number line: $\frac{1}{3}(2 x-1)<\frac{1}{4}(x+5)<\frac{1}{6}(3 x+4), x \in R$

Answer

$\frac{1}{3}(2 x-1)<\frac{1}{4}(x+5)$
$4(2 x-1)<3(x+5)$
$8 x-4<3 x+15$
$8 x-3 x<15+4$
$5 x<19$
$x<3 \frac{4}{5}$
and
$\frac{1}{4}(x+5)<\frac{1}{6}(3 x+4)$
$6(x+5)<4(3 x+4)$
$6 x+30<12 x+16$
$6 x-12 x<16-30$
$-6 x<-14$
$x>2 \frac{1}{3}$
Solution set $=[2 \frac{1}{3}]$

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Question 124 Marks

Solve the following linear in$-$equation and graph the solution set on a real number line: $4 \frac{3}{4} \geq x +\frac{5}{6}>\frac{1}{3}, x \in R$

Answer

$4 \frac{3}{4} \geq x+\frac{5}{6}$
$\frac{19}{4} \geq \frac{6 x+5}{6}$
$114 \geq 24 x+20$
$114-20 \geq 24 x$
$94 \geq 24 x$
$x \leq 3 \frac{11}{12}$
and
$x+\frac{5}{6}>\frac{1}{3}$
$\frac{6 x+5}{6}>\frac{1}{3}$
$18 x+15>6$
$18 x>6-15$
$18 x>-9$
$x>-\frac{1}{2}$
Solution set $=\left[-\frac{1}{2}< x \leq 3 \frac{11}{12}\right]$

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Question 134 Marks

Solve the following linear in$-$equation and graph the solution set on a real number line: $-3 \leq \frac{1}{2}-\frac{2 x }{3} \leq 2 \frac{2}{3}, x \in N$

Answer

$-3 \leq \frac{1}{2}-\frac{2 x }{3}$
$-3 \leq \frac{3-4 x }{6}$
$-18 \leq 3-4 x$
$-18-3 \leq-4 x$
$-21 \leq-4 x$
$x \leq \frac{21}{4}$
$x \leq 5 \frac{1}{4}$
and
$\frac{1}{2}-\frac{2 x}{3} \leq 2 \frac{2}{3}$
$\frac{3-4 x}{6} \leq \frac{8}{3}$
$9-12 x \leq 48$
$-12 x \leq 39$
$12 x \geq-39$
$x \geq-3 \frac{1}{4}$
Solution set $=\left[-3 \frac{1}{4} \leq x \leq 5 \frac{1}{4}\right]$

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Mathematics [4 marks sum]

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