Quadratic Inequality

How to solve a quadratic inequality: 3 examples and their solutions.

Example 1

Example

Solution

First, find the zeros of the left side.

Quadratic Function: Find Zeros

Factor the left side
x2 - 3x - 10.

Find a pair of numbers
whose product is the constant term -10
and whose sum is the coefficient of the middle term -3.

-5⋅2 = -10
-5 + 2 = -3

Then (x - 5)(x + 2) ≤ 0.

Factor a Quadratic Trinomial

Find the zeros.

Quadratic Equation: by Factoring

Case 1) x - 5 = 0
Then x = 5.

Case 2) x + 2 = 0
Then x = -2.

Case 1) x = 5
Case 2) x = -2

So the zeros are x = -2, 5.

Draw y = (x - 5)(x + 2)
on the x-axis.

First point the zeros x = -2 and 5.
And draw a parabola
that passes through x = -2 and 5.

Quadratic Function: Find Zeros

See (x - 5)(x + 2) ≤ 0.
The left side is less than or equal to 0.

So color the region
where the graph is below the x-axis (y = 0).

The inequality sign includes equal to [=].
So draw full circles on the zeros:
x = -2 and x = 5.

The colored region is
-2 ≤ x ≤ 5.

So
-2 ≤ x ≤ 5
is the answer.

Example 2

Example

Solution

First, find the zeros of the left side.

-16 = -42

Find the zeros.

Case 1) x + 4 = 0
Then x = -4.

Case 2) x - 4 = 0
Then x = 4.

Case 1) x = -4
Case 2) x = 4

So the zeros are x = ±4.

Draw y = (x + 4)(x - 4)
on the x-axis.

First point the zeros x = -4 and 4.
And draw a parabola
that passes through x = -4 and 4.

See (x + 4)(x - 4) ≤ 0.
The left side is greater than 0.

So color the regions
where the graph is above the x-axis (y = 0).

The inequality sign does not include equal to [=].
So draw empty circles on the zeros:
x = -4 and x = 4.

The colored regions are
x < -4 or x > 4.

So
x < -4 or x > 4
is the answer.

Example 3

Example

Solution

Divide both sides by (-).

Dividing both sides by (-)
does change the order of the inequality sign:
≥ → ≤.

Linear Inequality (One Variable)

Make a perfect square trinomial.

x2 is x2.

-10x is
-2 times
x times,
(-10x)/(-2⋅x), 5.

+25 is +52.

x2 - 2⋅x⋅5 + 52 = (x + 5)2

Find the zeros.

The zero is x = 5.

Draw y = (x + 5)2
on the x-axis.

First point the zero x = 5.
And draw a parabola
that touches x = 5.

See (x + 5)2 ≤ 0.

The left side is less than or equal to 0.

So color the region
where the graph is below the x-axis (y = 0).
But there's no region to color.

Then see the inequality sign.
The inequality sign includes equal to [=].
So draw a full circle on the zero:
x = 5.

The colored region is
x = 5.

So
x = 5
is the answer.