Solve x^2 + 3x - 4 >= 0 Step by Step | Quadratic inequalities

Step by step

The question

Question: solve x² + 3x − 4 ≥ 0.

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

x² + 3x − 4 ≥ 0

1 Find two numbers

We need two numbers.

They multiply to −4, and they add to 3.

? × ? = −4

? + ? = 3

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 4, smallest first.

We take the smaller from the bigger.

We need 3.

1 and 4 give 3.

Yes!

With the signs: −1 and 4.

1 × 4: 4 − 1 = 3 ✓

The two numbers go in the brackets.

So we get (x − 1)(x + 4) ≥ 0.

(x − 1)(x + 4) ≥ 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 + 4 = 0

In x − 1 = 0, we add 1 on both sides.

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

x − 1 + 1 = 0 + 1 x + 4 − 4 = 0 − 4

The critical points are 1 and −4.

x = 1 x = −4

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 −4 and 1 split the number line into 3 parts.

At −4, x + 4 is 0.

So the product is 0 there.

At 1, x − 1 is 0.

So the product is 0 there.

Left of −4, we test x = −5.

−5

x − 1 is −6 and x + 4 is −1.

−6

−1

Minus times minus is plus.

Between −4 and 1, we test x = 0.

x − 1 is −1 and x + 4 is 4.

−1

Minus times plus is minus.

Right of 1, we test x = 2.

x − 1 is 1 and x + 4 is 6.

Plus times plus is plus.

4 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 ≥ 1.

x ≤ −4 or x ≥ 1

≥ 0 → the + parts and the 0s

Careful!

Not the minus part!

Greater than 0 means plus.

Careful: not −4 ≤ x ≤ 1 ✗

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.

∞ means it goes on for ever.

∪ means or.

interval: (−∞, −4] ∪ [1, ∞)

6 Check

Let's check with a picture: the graph of y = x² + 3x − 4.

It is a parabola.

It crosses the x-axis at the critical points, −4 and 1.

Above the x-axis, y is plus.

That is x ≤ −4 or x ≥ 1: our answer.

Let's check: a number from the answer, 3, put into the question.

x = 3: 3² + 3(3) − 4 = 9 + 9 − 4 = 14

14 is greater than or equal to 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