StudentAidUnit 2: Quadratic Equations — Exercise 2 5
10th Class Mathematics · Unit 2: Quadratic Equations
2.5.1.(i) Solve the inequality x² + 3x - 4 > 0. (ii) Solve the inequality 2x² - 8x - 6 > 0. (iii) Solve the inequality x² + x - 6 < 0. (iv) Solve the inequality x² - 6x + 9 < 0. (v) Solve the inequality 4x² - 16x + 15 ≤ 0. (vi) Solve the inequality -x² + 3x - 2 ≥ 0.
Given
(i)
x2+3x-4>0
(i) Roots of x^2+3x-4=0
(x+4)(x-1)=0 ⇒ x=-4,1
(i) Need positive region
x<-4 or x>1
Result
(i)
(-infty,-4)cup(1,infty)
Given
(ii)
2x2-8x-6>0
(ii) Divide by 2, solve x^2-4x-3=0
x=frac{4 ± sqrt{28}{2}=2 ± sqrt7
(ii) Need positive region
x<2-sqrt7 or x>2+sqrt7
Result
(ii)
(-infty,2-sqrt7)cup(2+sqrt7,infty)
Given
(iii)
x2+x-6<0
(iii) Roots
(x+3)(x-2)=0 ⇒ x=-3,2
(iii) Need negative region
-3<x<2
Result
(iii)
(-3,2)
Given
(iv)
x2-6x+9<0
(iv)
(x-3)2<0
Result
(iv)
varnothing (no solution)
Given
(v)
4x2-16x+15le0
(v) Factor
(2x-5)(2x-3)=0 ⇒ x=frac52,frac32
(v) Need negative region incl. endpoints
frac32le xlefrac52
Result
(v)
left[frac32,frac52right]
Given
(vi)
-x2+3x-2ge0
(vi) Multiply by -1 (reverse inequality)
x2-3x+2le0 ⇒ (x-1)(x-2)=0
(vi) Need negative region incl. endpoints
1le xle2
Result
(vi)
[1,2]