Solve x^2 + x >= 12 Step by Step | Quadratic inequalities

Step by step

The question

Question: solve x² + x ≥ 12.

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

x² + x ≥ 12

1 Get 0 on one side

We want 0 on one side.

So we take 12 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 + x − 12 ≥ 12 − 12

Now we have x² + x − 12 ≥ 0.

x² + x − 12 ≥ 0

2 Find two numbers

We need two numbers.

They multiply to −12, and they add to 1.

? × ? = −12

? + ? = 1

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 12, smallest first.

We take the smaller from the bigger.

We need 1.

3 and 4 give 1.

Yes!

With the signs: −3 and 4.

1 × 12: 12 − 1 = 11

2 × 6: 6 − 2 = 4

3 × 4: 4 − 3 = 1 ✓

The two numbers go in the brackets.

So we get (x − 3)(x + 4) ≥ 0.

(x − 3)(x + 4) ≥ 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 + 4 = 0

In x − 3 = 0, we add 3 on both sides.

In x + 4 = 0, we take away 4 on both sides.

x − 3 + 3 = 0 + 3 x + 4 − 4 = 0 − 4

The critical points are 3 and −4.

x = 3 x = −4

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 −4 and 3 split the number line into 3 parts.

At −4, x + 4 is 0.

So the product is 0 there.

At 3, x − 3 is 0.

So the product is 0 there.

Left of −4, we test x = −5.

−5

x − 3 is −8 and x + 4 is −1.

−8

−1

Minus times minus is plus.

Between −4 and 3, we test x = 0.

x − 3 is −3 and x + 4 is 4.

−3

Minus times plus is minus.

Right of 3, we test x = 4.

x − 3 is 1 and x + 4 is 8.

Plus times plus is plus.

5 Read the answer

Greater than or equal to 0 means positive or 0.

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

So x ≤ −4 or x ≥ 3.

x ≤ −4 or x ≥ 3

≥ 0 → the + parts and the 0s

Careful!

Not the minus part!

Greater than 0 means plus.

Careful: not −4 ≤ x ≤ 3 ✗

6 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.

∞ means it goes on for ever.

∪ means or.

interval: (−∞, −4] ∪ [3, ∞)

7 Check

Let's check with a picture: the graph of y = x² + x − 12.

It is a parabola.

It crosses the x-axis at the critical points, −4 and 3.

Above the x-axis, y is plus.

That is x ≤ −4 or x ≥ 3: our answer.

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

x = 5: 5² + 5 = 25 + 5 = 30

30 is greater than or equal to 12.

True!

30 ≥ 12 true

Let's look back.

First: get 0 on one side.

Then: 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