Solve x^2 < 9 Step by Step | Quadratic inequalities

Step by step

The question

Question: solve x² < 9.

There is an x², so the answer can be two parts.

x² < 9

1 Get 0 on one side

We want 0 on one side.

So we take 9 away from both sides.

We do it on both sides.

Why both sides?

The sign compares the two sides.

If we change only one side, the answer can change.

x² − 9 < 9 − 9

Now we have x² − 9 < 0.

x² − 9 < 0

2 Two squares

This is one square take away another square.

x² is x squared.

9 is 3 squared.

A square minus a square is the plus bracket times the minus bracket.

a² − b² = (a + b)(a − b)

x² = (x)²

9 = 3²

So we get (x + 3)(x − 3) < 0.

(x + 3)(x − 3) < 0

3 Find the critical points

A critical point is an x where the side with x is 0, or has no value.

Only there can its sign change.

critical point: where it is 0 (or ÷ 0)

We make each bracket 0.

x + 3 = 0 x − 3 = 0

In x + 3 = 0, we take away 3 on both sides.

In x − 3 = 0, we add 3 on both sides.

x + 3 − 3 = 0 − 3 x − 3 + 3 = 0 + 3

The critical points are −3 and 3.

x = −3 x = 3

4 Make a sign table

Between the critical points the sign stays the same.

So one test number tells us the sign of each part.

We draw a table.

The critical points −3 and 3 split the number line into 3 parts.

At −3, x + 3 is 0.

So the product is 0 there.

At 3, x − 3 is 0.

So the product is 0 there.

Left of −3, we test x = −4.

−4

x + 3 is −1 and x − 3 is −7.

−1

−7

Minus times minus is plus.

Between −3 and 3, we test x = 0.

x + 3 is 3 and x − 3 is −3.

−3

Plus times minus is minus.

Right of 3, we test x = 4.

x + 3 is 7 and x − 3 is 1.

Plus times plus is plus.

5 Read the answer

Less than 0 means negative.

So we want the parts with a minus.

So −3 < x < 3.

−3 < x < 3

< 0 → the − parts

Careful!

Not the plus parts!

Less than 0 means minus.

Careful: not x < −3 or x > 3 ✗

6 Show it on a number line

Books also write the answer like this.

It is called interval notation.

A round bracket means the end is not in, like the open circle.

interval: (−3, 3)

7 Check

Let's check with a picture: the graph of y = x² − 9.

It is a parabola.

It crosses the x-axis at the critical points, −3 and 3.

Below the x-axis, y is minus.

That is −3 < x < 3: our answer.

Let's check: a number from the answer, 0, put into the question.

x = 0: 0² = 0

0 is less than 9.

True!

0 < 9 true

Let's look back.

First: get 0 on one side.

Then: two squares.

Then: find the critical points.

Then: make a sign table.

Then: read the answer.

Then: show it on a number line.

Last: check.

Answer