Answer to Question #177488 in Calculus for Talha

Question

Q1. Solve the inequalities and show the solution sets on the real line.

a) −2𝑥2−𝑥+3≥0

b) 13𝑥+23<56𝑥−1

c) 3𝑥−3≥𝑥−5

d) 4<𝑥2<9

e) √(𝑥−1)2 <√4

Answer 1

a)

-2x²-x+3≥0

=> 2x²+x-3≤0

=> 2x²+3x-2x-3≤0

=> (2x+3)(x-1)≤0

=> (x+1.5)(x-1)≤0

So x $\isin$ [-1.5, 1]

b)

13x+23 < 56x-1

=> 13x-56x < -1-23

=> -43x < -24

=> x > 24/43

So x $\isin ( \frac{24}{43} , ∞)$

c)

3x-3≥x-5

=> 3x - x ≥ -5+3

=> 2x ≥ -2

=> x ≥ -1

So x $\isin [-1, ∞)$

d)

4 < x² < 9

=> x² > 4 and x² <9

=> {x <-2 or x > 2 } and -3<x<3

=> {-3<x<-2} or {2<x<3}

So x $\isin (-3,-2)\bigcup(2,3)$

e)

$\sqrt{(x-1)²} < \sqrt{4}$

=> | x - 1 | < 2

=> -2 < x - 1 < 2

=> -2 +1 < x < 2+1

=> -1 < x < 3

So x $\isin (-1, 3)$