# A Guidline for Solving Quadratic Inequalities

Here is a simple procedure you might want to follow.

1. If the coefficient of x^{2} is negative, multiply −1 to everything to

make the leading coefficient positive. When you do this, remember

that the direction of the inequality changes.

Example)

−x^{2} + 3x + 4 < 0

is changed to:

x^{2} − 3x − 4 > 0

2. Factor the polynomial and write down the roots.

Example)

x^{2 } − 3x − 4 > 0

when factored, you get

(x − 4)(x + 1) > 0

So the two roots are x = 4 and x = −1.

3. If the inequality is < or ≤ then x is in between the roots. If not,

x is bigger than the larger root or less then the smaller root.

Example)

(x − 4)(x + 1) > 0

has solution : x > 4 or x < −1.

While

(x − 4)(x + 1) < 0

has solution : −1 < x < 4.

Also

(x − 4)(x + 1) ≤ 0

has solution : −1≤ x ≤4.

Some more examples:

a.

3x^{2} − 4x − 7 < 0

(3x − 7)(x + 1) < 0

roots: x = 7/3 , −1.

The answer is : −1 < x < 7/3 .

b.

−x^{2} − 6x − 5 < 0

x^{2} + 6x + 5 > 0

(x + 1)(x + 5) > 0

roots: x = −5 , −1.

The answer is : x > −1 or x < −5.

c.

−x^{2} + 9x − 14 ≥0

x^{2} − 9x + 14 ≤0

(x − 7)(x − 2) ≤0

roots: x = 2 , 7

The answer is : 2 ≤x ≤7.