[5 marks sum]

Question 15 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 25 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 35 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

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

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Question 45 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} $
$ \text { 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 55 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 [5 marks sum]

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