Solve 2x^2 - 8 > 0 Step by Step | Quadratic inequalities

Step by step

The question

Question: solve 2x² − 8 > 0.

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

2x² − 8 > 0

Every number can be divided by 2.

We divide both sides by 2.

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

0 divided by 2 is still 0.

x² − 4 > 0

1 Two squares

This is one square take away another square.

x² is x squared.

4 is 2 squared.

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

a² − b² = (a + b)(a − b)

x² = (x)²

4 = 2²

So we get (x + 2)(x − 2) > 0.

(x + 2)(x − 2) > 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 − 2 = 0

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

In x − 2 = 0, we add 2 on both sides.

x + 2 − 2 = 0 − 2 x − 2 + 2 = 0 + 2

The critical points are −2 and 2.

x = −2 x = 2

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 2 split the number line into 3 parts.

At −2, x + 2 is 0.

So the product is 0 there.

At 2, x − 2 is 0.

So the product is 0 there.

Left of −2, we test x = −3.

−3

x + 2 is −1 and x − 2 is −5.

−1

−5

Minus times minus is plus.

Between −2 and 2, we test x = 0.

x + 2 is 2 and x − 2 is −2.

−2

Plus times minus is minus.

Right of 2, we test x = 3.

x + 2 is 5 and x − 2 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 > 2.

x < −2 or x > 2

> 0 → the + parts

Careful!

Not the minus part!

Greater than 0 means plus.

Careful: not −2 < x < 2 ✗

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) ∪ (2, ∞)

6 Check

Let's check with a picture: the graph of y = x² − 4.

It is a parabola.

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

Above the x-axis, y is plus.

That is x < −2 or x > 2: our answer.

Let's check: a number from the answer, 4, put into the question.

x = 4: 2(4)² − 8 = 32 − 8 = 24

24 is greater than 0.

True!

24 > 0 true

Let's look back.

First: 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