Solve 3x^2 + 6x - 9 < 0 Step by Step | Quadratic inequalities

Step by step

The question

Question: solve 3x² + 6x − 9 < 0.

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

3x² + 6x − 9 < 0

Every number can be divided by 3.

We divide both sides by 3.

3 is a positive number, so the sign stays the same.

0 divided by 3 is still 0.

x² + 2x − 3 < 0

1 Find two numbers

We need two numbers.

They multiply to −3, and they add to 2.

? × ? = −3

? + ? = 2

They multiply to a minus number, so one number is plus and one is minus.

They add to a plus number, so the bigger one is plus.

× is − → one +, one −; bigger one +

We list the pairs that make 3, smallest first.

We take the smaller from the bigger.

We need 2.

1 and 3 give 2.

Yes!

With the signs: −1 and 3.

1 × 3: 3 − 1 = 2 ✓

The two numbers go in the brackets.

So we get (x − 1)(x + 3) < 0.

(x − 1)(x + 3) < 0

2 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 − 1 = 0 x + 3 = 0

In x − 1 = 0, we add 1 on both sides.

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

x − 1 + 1 = 0 + 1 x + 3 − 3 = 0 − 3

The critical points are 1 and −3.

x = 1 x = −3

3 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 1 split the number line into 3 parts.

At −3, x + 3 is 0.

So the product is 0 there.

At 1, x − 1 is 0.

So the product is 0 there.

Left of −3, we test x = −4.

−4

x − 1 is −5 and x + 3 is −1.

−5

−1

Minus times minus is plus.

Between −3 and 1, we test x = 0.

x − 1 is −1 and x + 3 is 3.

−1

Minus times plus is minus.

Right of 1, we test x = 2.

x − 1 is 1 and x + 3 is 5.

Plus times plus is plus.

4 Read the answer

Less than 0 means negative.

So we want the parts with a minus.

So −3 < x < 1.

−3 < x < 1

< 0 → the − parts

Careful!

Not the plus parts!

Less than 0 means minus.

Careful: not x < −3 or x > 1 ✗

5 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, 1)

6 Check

Let's check with a picture: the graph of y = x² + 2x − 3.

It is a parabola.

It crosses the x-axis at the critical points, −3 and 1.

Below the x-axis, y is minus.

That is −3 < x < 1: our answer.

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

x = 0: 3(0)² + 6(0) − 9 = 0 + 0 − 9 = −9

−9 is less than 0.

True!

−9 < 0 true

Let's look back.

First: find two numbers.

Then: find the critical points.

Then: make a sign table.

Then: read the answer.

Then: show it on a number line.

Last: check.

Answer