Solve x^2 - x - 6 > 0 Step by Step | Quadratic inequalities

Step by step

The question

Question: solve x² − x − 6 > 0.

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

x² − x − 6 > 0

1 Find two numbers

We need two numbers.

They multiply to −6, and they add to −1.

? × ? = −6

? + ? = −1

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

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

× is − → one +, one −; bigger one −

We list the pairs that make 6, smallest first.

We take the smaller from the bigger.

We need 1.

2 and 3 give 1.

Yes!

With the signs: 2 and −3.

1 × 6: 6 − 1 = 5

2 × 3: 3 − 2 = 1 ✓

The two numbers go in the brackets.

So we get (x + 2)(x − 3) > 0.

(x + 2)(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 + 2 = 0 x − 3 = 0

In x + 2 = 0, we take away 2 on both sides.

In x − 3 = 0, we add 3 on both sides.

x + 2 − 2 = 0 − 2 x − 3 + 3 = 0 + 3

The critical points are −2 and 3.

x = −2 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 −2 and 3 split the number line into 3 parts.

At −2, x + 2 is 0.

So the product is 0 there.

At 3, x − 3 is 0.

So the product is 0 there.

Left of −2, we test x = −3.

−3

x + 2 is −1 and x − 3 is −6.

−1

−6

Minus times minus is plus.

Between −2 and 3, we test x = 0.

x + 2 is 2 and x − 3 is −3.

−3

Plus times minus is minus.

Right of 3, we test x = 4.

x + 2 is 6 and x − 3 is 1.

Plus times plus is plus.

4 Read the answer

Greater than 0 means positive.

So we want the parts with a plus.

So x < −2 or x > 3.

x < −2 or x > 3

> 0 → the + parts

Careful!

Not the minus part!

Greater than 0 means plus.

Careful: not −2 < x < 3 ✗

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.

∞ means it goes on for ever.

∪ means or.

interval: (−∞, −2) ∪ (3, ∞)

6 Check

Let's check with a picture: the graph of y = x² − x − 6.

It is a parabola.

It crosses the x-axis at the critical points, −2 and 3.

Above the x-axis, y is plus.

That is x < −2 or x > 3: our answer.

Let's check: a number from the answer, 5, put into the question.

x = 5: 5² − 5 − 6 = 25 − 5 − 6 = 14

14 is greater than 0.

True!

14 > 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