Class 11: Linear In-equations – Multiple Choice uestions

$\displaystyle \textbf{Question 1. }\text{If }x<7,\text{ then}$
$\displaystyle \text{(a) }-x<-7\qquad \text{(b) }-x\leq-7\qquad \text{(c) }-x>-7\qquad \text{(d) }-x\geq-7$
$\displaystyle \text{Answer:}$
$\displaystyle x<7$
$\displaystyle \text{Multiplying both sides by }-1\text{, the inequality sign reverses.}$
$\displaystyle -x>-7$
$\displaystyle \therefore \text{Correct option is (c).}$
$\\$

$\displaystyle \textbf{Question 2. }\text{If }-3x+17<-13,\text{ then}$
$\displaystyle \text{(a) }x\in(10,\infty)\qquad \text{(b) }x\in[10,\infty)\qquad \text{(c) }x\in(-\infty,10]\qquad \text{(d) }x\in[-10,10)$
$\displaystyle \text{Answer:}$
$\displaystyle -3x+17<-13$
$\displaystyle -3x<-30$
$\displaystyle x>10$
$\displaystyle \therefore x\in(10,\infty)$
$\displaystyle \therefore \text{Correct option is (a).}$
$\\$

$\displaystyle \textbf{Question 3. }\text{Given that }x,y\text{ and }b\text{ are real numbers and }x<y,\ b>0,\text{ then}$
$\displaystyle \text{(a) }\frac{x}{b}<\frac{y}{b}\qquad \text{(b) }\frac{x}{b}\leq\frac{y}{b}\qquad \text{(c) }\frac{x}{b}>\frac{y}{b}\qquad \text{(d) }\frac{x}{b}\geq\frac{y}{b}$
$\displaystyle \text{Answer:}$
$\displaystyle x<y,\quad b>0$
$\displaystyle \text{Dividing both sides by the positive number }b\text{, the sign remains unchanged.}$
$\displaystyle \frac{x}{b}<\frac{y}{b}$
$\displaystyle \therefore \text{Correct option is (a).}$
$\\$

$\displaystyle \textbf{Question 4. }\text{If }x\text{ is a real number and }|x|<5,\text{ then}$
$\displaystyle \text{(a) }x\geq5\qquad \text{(b) }-5<x<5\qquad \text{(c) }x\leq-5\qquad \text{(d) }-5\leq x\leq5$
$\displaystyle \text{Answer:}$
$\displaystyle |x|<5$
$\displaystyle \Rightarrow -5<x<5$
$\displaystyle \therefore \text{Correct option is (b).}$
$\\$

$\displaystyle \textbf{Question 5. }\text{If }x\text{ and }a\text{ are real numbers such that }a>0\text{ and }|x|>a,\text{ then}$
$\displaystyle \text{(a) }x\in(-a,\infty)\qquad \text{(b) }x\in[-\infty,a]\qquad \\ \text{(c) }x\in(-a,a)\qquad \text{(d) }x\in(-\infty,-a)\cup(a,\infty)$
$\displaystyle \text{Answer:}$
$\displaystyle |x|>a,\quad a>0$
$\displaystyle \Rightarrow x<-a\text{ or }x>a$
$\displaystyle \therefore x\in(-\infty,-a)\cup(a,\infty)$
$\displaystyle \therefore \text{Correct option is (d).}$
$\\$

$\displaystyle \textbf{Question 6. }\text{If }|x-1|>5,\text{ then}$
$\displaystyle \text{(a) }x\in(-4,6)\qquad \text{(b) }x\in[-4,6]\qquad \\ \text{(c) }x\in(-\infty,-4)\cup(6,\infty)\qquad \text{(d) }x\in(-\infty,-4)\cup[6,\infty)$
$\displaystyle \text{Answer:}$
$\displaystyle |x-1|>5$
$\displaystyle x-1<-5\text{ or }x-1>5$
$\displaystyle x<-4\text{ or }x>6$
$\displaystyle \therefore x\in(-\infty,-4)\cup(6,\infty)$
$\displaystyle \therefore \text{Correct option is (c).}$
$\\$

$\displaystyle \textbf{Question 7. }\text{If }|x+2|\leq9,\text{ then}$
$\displaystyle \text{(a) }x\in(-7,11)\qquad \text{(b) }x\in[-11,7]\qquad \\ \text{(c) }x\in(-\infty,-7)\cup(11,\infty)\qquad \text{(d) }x\in(-\infty,-7)\cup[11,\infty)$
$\displaystyle \text{Answer:}$
$\displaystyle |x+2|\leq9$
$\displaystyle -9\leq x+2\leq9$
$\displaystyle -11\leq x\leq7$
$\displaystyle \therefore x\in[-11,7]$
$\displaystyle \therefore \text{Correct option is (b).}$
$\\$

$\displaystyle \textbf{Question 8. }\text{The inequality representing the following graph is}$
$\displaystyle \text{(a) }|x|<3\qquad \text{(b) }|x|\leq3\qquad \text{(c) }|x|>3\qquad \text{(d) }|x|\geq3$
$\displaystyle \text{Answer:}$
$\displaystyle \text{The graph represents all points between }-3\text{ and }3\text{ including the end points.}$
$\displaystyle \therefore -3\leq x\leq3$
$\displaystyle \therefore |x|\leq3$
$\displaystyle \therefore\text{Correct option is (b).}$
$\\$

$\displaystyle \textbf{Question 9. }\text{The linear inequality representing the solution set given in is}$
$\displaystyle \text{(a) }|x|<5\qquad \text{(b) }|x|>5\qquad \text{(c) }|x|\geq5\qquad \text{(d) }|x|\leq5$
$\displaystyle \text{Answer:}$
$\displaystyle \text{The graph represents all points outside the interval }[-5,5]\text{ including the end points.}$
$\displaystyle \therefore x\leq-5\quad \text{or}\quad x\geq5$
$\displaystyle \therefore |x|\geq5$
$\displaystyle \therefore\text{Correct option is (c).}$
$\\$

$\displaystyle \textbf{Question 10. }\text{The solution set of the inequation }|x+2|\leq5\text{ is}$
$\displaystyle \text{(a) }(-7,5)\qquad \text{(b) }[-7,3]\qquad \text{(c) }[-5,5]\qquad \text{(d) }(-7,3)$
$\displaystyle \text{Answer:}$
$\displaystyle |x+2|\leq5$
$\displaystyle -5\leq x+2\leq5$
$\displaystyle -7\leq x\leq3$
$\displaystyle \therefore x\in[-7,3]$
$\displaystyle \therefore \text{Correct option is (b).}$
$\\$

$\displaystyle \textbf{Question 11. }\text{If }\frac{|x-2|}{x-2}\geq0,\text{ then}$
$\displaystyle \text{(a) }x\in[2,\infty)\qquad \text{(b) }x\in(2,\infty)\qquad \text{(c) }x\in(-\infty,2)\qquad \text{(d) }x\in(-\infty,2]$
$\displaystyle \text{Answer:}$
$\displaystyle \frac{|x-2|}{x-2}\geq0,\quad x\neq2$
$\displaystyle \text{If }x>2,\text{ then }|x-2|=x-2$
$\displaystyle \frac{|x-2|}{x-2}=1\geq0$
$\displaystyle \text{If }x<2,\text{ then }|x-2|=-(x-2)$
$\displaystyle \frac{|x-2|}{x-2}=-1<0$
$\displaystyle \therefore x\in(2,\infty)$
$\displaystyle \therefore \text{Correct option is (b).}$
$\\$

$\displaystyle \textbf{Question 12. }\text{If }|x+3|\geq10,\text{ then}$
$\displaystyle \text{(a) }x\in(-13,7]\qquad \text{(b) }x\in(-13,7)\qquad \\ \text{(c) }x\in(-\infty,-13)\cup(7,\infty)\qquad \text{(d) }x\in(-\infty,-13]\cup[7,\infty)$
$\displaystyle \text{Answer:}$
$\displaystyle |x+3|\geq10$
$\displaystyle x+3\leq-10\text{ or }x+3\geq10$
$\displaystyle x\leq-13\text{ or }x\geq7$
$\displaystyle \therefore x\in(-\infty,-13]\cup[7,\infty)$
$\displaystyle \therefore \text{Correct option is (d).}$
$\\$