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

Step by step

The question

Question: solve x² − 5x + 6 < 0.

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

x² − 5x + 6 < 0

1 Find two numbers

We need two numbers.

They multiply to 6, and they add to −5.

? × ? = 6

? + ? = −5

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

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

× is + → same signs: both −

We list the pairs that make 6, smallest first.

We add each pair.

We need 5.

2 and 3 give 5.

Yes!

With the signs: −2 and −3.

1 × 6: 1 + 6 = 7

2 × 3: 2 + 3 = 5 ✓

The two numbers go in the brackets.

So we get (x − 2)(x − 3) < 0.

(x − 2)(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 − 2 = 0 x − 3 = 0

In x − 2 = 0, we add 2 on both sides.

In x − 3 = 0, we add 3 on both sides.

x − 2 + 2 = 0 + 2 x − 3 + 3 = 0 + 3

The critical points are 2 and 3.

x = 2 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 2 and 3 split the number line into 3 parts.

At 2, x − 2 is 0.

So the product is 0 there.

At 3, x − 3 is 0.

So the product is 0 there.

Left of 2, we test x = 0.

x − 2 is −2 and x − 3 is −3.

−2

−3

Minus times minus is plus.

Between 2 and 3, we test x = 5/2.

x − 2 is 1/2 and x − 3 is −1/2.

−12

Plus times minus is minus.

Right of 3, we test x = 4.

x − 2 is 2 and x − 3 is 1.

Plus times plus is plus.

4 Read the answer

Less than 0 means negative.

So we want the parts with a minus.

So 2 < x < 3.

2 < x < 3

< 0 → the − parts

Careful!

Not the plus parts!

Less than 0 means minus.

Careful: not x < 2 or x > 3 ✗

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.

interval: (2, 3)

6 Check

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

It is a parabola.

It crosses the x-axis at the critical points, 2 and 3.

Below the x-axis, y is minus.

That is 2 < x < 3: our answer.

Let's check: a number from the answer, 5/2, put into the question.

x = 5 over 2: (52)² − 5(52) + 6

= 254 − 252 + 6 = −14

−1/4 is less than 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