Solve x^2 + 6x + 5 <= 0 Step by Step | Quadratic inequalities

Step by step

The question

Question: solve x² + 6x + 5 ≤ 0.

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

x² + 6x + 5 ≤ 0

1 Find two numbers

We need two numbers.

They multiply to 5, and they add to 6.

? × ? = 5

? + ? = 6

They multiply to a plus number, so the two numbers have the same sign.

They add to a plus number, so both are plus.

× is + → same signs: both +

We list the pairs that make 5, smallest first.

We add each pair.

We need 6.

1 and 5 give 6.

Yes!

1 × 5: 1 + 5 = 6 ✓

The two numbers go in the brackets.

So we get (x + 1)(x + 5) ≤ 0.

(x + 1)(x + 5) ≤ 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 + 5 = 0

In x + 1 = 0, we take away 1 on both sides.

In x + 5 = 0, we take away 5 on both sides.

x + 1 − 1 = 0 − 1 x + 5 − 5 = 0 − 5

The critical points are −1 and −5.

x = −1 x = −5

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 −5 and −1 split the number line into 3 parts.

At −5, x + 5 is 0.

So the product is 0 there.

At −1, x + 1 is 0.

So the product is 0 there.

Left of −5, we test x = −6.

−6

x + 1 is −5 and x + 5 is −1.

−5

−1

Minus times minus is plus.

Between −5 and −1, we test x = −4.

−4

x + 1 is −3 and x + 5 is 1.

−3

Minus times plus is minus.

Right of −1, we test x = 0.

x + 1 is 1 and x + 5 is 5.

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 −5 ≤ x ≤ −1.

−5 ≤ x ≤ −1

≤ 0 → the − parts and the 0s

Careful!

Not the plus parts!

Less than 0 means minus.

Careful: not x ≤ −5 or 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.

interval: [−5, −1]

6 Check

Let's check with a picture: the graph of y = x² + 6x + 5.

It is a parabola.

It crosses the x-axis at the critical points, −5 and −1.

Below the x-axis, y is minus.

That is −5 ≤ x ≤ −1: our answer.

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

x = −4: (−4)² + 6(−4) + 5 = 16 − 24 + 5

= −3

−3 is less than or equal to 0.

True!

−3 ≤ 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