[4 marks sum]

Question 14 Marks

Given: P {x : 5 < 2x – 1 ≤ 11, x∈R)
Q{x : – 1 ≤ 3 + 4x < 23, x∈I) where
R = (real numbers), I = (integers)
Represent P and Q on number line. Write down the elements of P ∩ Q.

Answer

P = {x : 5 < 2x – 1 ≤ 11}
5 < 2x – 1 ≤ 11
5 < 2 x – 1 and 2x – 1 ≤ 11
– 2 x < – 5 – 1 and 2 x ≤ 11 + 1
– 2x < – 6 and 2x ≤ 12
–x < –3
x > 3 or 3 < x
∴ Solution set = 3 < x ≤ 6 - {4, 5, 6}
Solution set on number line.

Q = {–1 ≤ 3 + 4x < 23}
–1 ≤ 3 + 4 x < 23
–1 < 3 + 4x and 3 + 4 x < 23
–4x < 3 + 1 and 4x < 23 - 3
–4x < 4 and 4x < 20
–x < 1 and x < 5
x > – 1
–1 < x
∴ Solution set = {0, 1, 2, 3, 4}
∴ Solution set on number line

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Question 24 Marks

Solve the following inequation and represent the solution set on the number line:
$
-3<-\frac{1}{2}-\frac{2 x}{3} \leq \frac{5}{6}, x \in R
$

Answer

$
-3<-\frac{1}{2}-\frac{2 x}{3} \leq \frac{5}{6}, x \in R
$
(i) $-3<-\frac{1}{2}-\frac{2 x}{3}$
$\Rightarrow-3 \leq-\left(\frac{1}{2}+\frac{2 x}{3}\right)$
$\Rightarrow-\left(\frac{1}{2}+\frac{2 x}{3}\right)>-3$
$\Rightarrow-\frac{2 x}{3}>-3+\frac{1}{2}$
$\Rightarrow-\frac{2 x}{3}>\frac{-5}{2}$
$\Rightarrow \frac{2 x^3}{3}<\frac{5}{2}$
$\Rightarrow x<\frac{5}{2} \times \frac{3}{2}$
$\Rightarrow x<\frac{15}{4}....(1)$
(ii) $-\frac{1}{2}-\frac{2 x}{3} \leq \frac{5}{6}$
$\Rightarrow-\frac{2 x}{3} \leq \frac{5}{6}+\frac{1}{2}$
$\Rightarrow \frac{-2 x}{3} \leq \frac{5+3}{6}$
$\Rightarrow \frac{-2}{3} \times \leq \frac{8}{6}$
$\Rightarrow \frac{2}{3} x \geq \frac{-8}{6}$
$\Rightarrow x \geq \frac{-8}{6} \times \frac{3}{2}$
$\Rightarrow x \geq-2$
$\Rightarrow-2 \leq x....(2)$
$\Rightarrow$ From (i) and (ii),
$-2 \leq \leq \frac{15}{4}$
$\therefore$ Solution $=\left\{x: x \in R ,-2 \leq x<\frac{15}{4}\right\}$
Now solution on number line

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Mathematics [4 marks sum]

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