Solve -x^2 + 9 >= 0 Step by Step | Quadratic inequalities

Step by step

The question

Question: solve −x² + 9 ≥ 0.

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

−x² + 9 ≥ 0

We want x² to be plus.

So we times both sides by −1.

Times a negative number turns the sign around.

Greater than or equal to becomes less than or equal to.

x² − 9 ≤ 0

1 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

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 + 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

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 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.

4 Read the answer

Less than or equal to 0 means negative or 0.

So we want the minus parts, and the 0s.

So −3 ≤ x ≤ 3.

−3 ≤ x ≤ 3

≤ 0 → the − parts and the 0s

Careful!

Not the plus parts!

Less than 0 means minus.

Careful: not x ≤ −3 or x ≥ 3 ✗

5 Show it on a number line

Books also write the answer like this.

It is called interval notation.

A square bracket means the end is in, like the filled circle.

interval: [−3, 3]

6 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: −(0)² + 9 = 0 + 9 = 9

9 is greater than or equal to 0.

True!

9 ≥ 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