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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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# A Guidline for Solving Quadratic Inequalities

Here is a simple procedure you might want to follow.

1. If the coefficient of x2 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)

−x2 + 3x + 4 < 0

is changed to:

x2 − 3x − 4 > 0

2. Factor the polynomial and write down the roots.
Example)

x2 − 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.

3x2 − 4x − 7 < 0
(3x − 7)(x + 1) < 0

roots: x = 7/3 , −1.

The answer is : −1 < x < 7/3 .

b.
−x2 − 6x − 5 < 0
x2 + 6x + 5 > 0
(x + 1)(x + 5) > 0

roots: x = −5 , −1.

The answer is : x > −1 or x < −5.
c.
−x2 + 9x − 14 ≥0
x2 − 9x + 14 ≤0
(x − 7)(x − 2) ≤0

roots: x = 2 , 7
The answer is : 2 ≤x ≤7.