$\mathbf{4}|x-1| \leq 5,|x| \geq 2 $
$$\begin{array}{lr} & |x-1| \leq 5 \\ & -5 \leq x-1 \leq 5 \\ \Rightarrow & -4 \leq x \leq 6 \\ \Rightarrow & x \in[-4,6]\quad \supset\text{(i)} \end{array}$$
$$ \begin{array}{lr} \text { and } & |x| \geq 2 \\ \Rightarrow & x \leq-2 \text { or } x \geq 2 \\ \Rightarrow & x \in(-\infty,-2] \cup[2, \infty)\quad \supset\text{(ii)} \end{array}$$
On combining Eqs. (i) and (ii), we get
$x\in(-4,-2]\cup [2,6]$
$-5 \leq \frac{2-3 x}{4} \leq 9$
$$\begin{aligned} &\begin{aligned} \text { We have, } \quad & -5 \leq \frac{2-3 x}{4} \\ \Rightarrow \quad& -20 \leq 2-3 x \quad \text { [multiplying by } 4 \text { on both sides] }\\ \Rightarrow \quad& 3 x \leq 2+20 \\ \Rightarrow \quad& 3 x \leq 22 \\ \Rightarrow \quad& x \leq \frac{22}{3} \\ \text { and }\quad & \frac{2-3 x}{4} \leq 9 \end{aligned}\\ \end{aligned}$$
$$\begin{array}{lc} \Rightarrow & 2-3 x \leq 36 \\ \Rightarrow & -3 x \leq 36-2 \\ \Rightarrow & -3 x \leq 34 \\ \Rightarrow & 3 x \geq-34 \\ \Rightarrow & x \geq-\frac{34}{3} \\ \Rightarrow & -\frac{34}{3} \leq x \leq \frac{22}{3} \\ \Rightarrow & x \in\left[\frac{-34}{3}, \frac{22}{3}\right] \end{array}$$
$4 x+3 \geq 2 x+17,3 x-5<-2$
We have,
$$\begin{aligned} & 4 x+3 \geq 2 x+17 \\ \Rightarrow \quad & 4 x-2 x \geq 17-3 \Rightarrow 2 x \geq 14 \\ \Rightarrow \quad & x \geq \frac{14}{2} \\ \Rightarrow \quad & x \geq 7 \quad \supset\text{(i)}\\ \text { Also, we have } & \\ & 3 x-5<-2 \\ \Rightarrow\quad & 3 x<-2+5 \Rightarrow 3 x<3 \\ \Rightarrow \quad & x<1\quad \supset\text{(ii)} \end{aligned}$$
On combining Eqs. (i) and (ii), we see that solution is not possible because nothing is common between these two solutions. (i.e., $x<1, x \geq 7$ ).
A company manufactures cassettes. Its cost and revenue functions are $C(x)=26000+30 x$ and $R(x)=43 x$, respectively, where $x$ is the number of cassettes produced and sold in a week. How many cassettes must be sold by the company to realise some profit?
$$\begin{aligned} \text{Cost function, }\quad C(x) =26000+30 x \\ \text{and revenue function, }\quad R(x) =43 x \\ \text{For profit,}\quad R(x) >C(x) \\ \Rightarrow \quad 26000+30 x <43 x \\ \Rightarrow \quad 30 x-43 x <-26000 \\ \Rightarrow \quad -13 x <-26000 \\ \Rightarrow \quad 13 x >26000 \\ \Rightarrow \quad x >\frac{26000}{13} \\ \therefore \quad x >2000 \end{aligned}$$
Hence, more than 2000 cassettes must be produced to get profit.
The water acidity in a pool is considered normal when the average pH reading of three daily measurements is between 8.2 and 8.5 . If the first two pH readings are 8.48 and 8.35 , then find the range of pH value for the third reading that will result in the acidity level being normal.
Given, $\quad\text{ first pH value} = 8.48$
and $\quad \text{second pH value}$ = 8.35
Let third pH value be $x$.
Since, it is given that average pH value lies between 8.2 and 8.5 .
$$\begin{aligned} & \therefore \quad 8.2< \frac{8.48+8.35+x}{3}< 8.5 \\ & \Rightarrow \quad 8.2 < \frac{16.83+x}{3} < 8.5 \\ & \Rightarrow \quad 3 \times 8.2 < 16.83+x < 8.5 \times 3 \\ & \Rightarrow \quad 24.6 < 16.83+x < 25.5 \\ & \Rightarrow \quad 24.6-16.83 < x< 25.5-16.83 \\ & \Rightarrow \quad 7.77< x<8.67 \end{aligned}$$ Thus, third pH value lies between 7.77 and 8.67 .