Solve 2x^2 - 7x + 3 < 0 Step by Step | Quadratic inequalities

Step by step

The question

Question: solve 2x² − 7x + 3 < 0.

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

2x² − 7x + 3 < 0

1 Find two numbers

The number in front of x² is 2.

So we multiply 2 by 3: that is 6.

We need two numbers that multiply to 6 and add to −7.

2 × 3 = 6

? × ? = 6, ? + ? = −7

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 7.

1 and 6 give 7.

Yes!

With the signs: −1 and −6.

1 × 6: 1 + 6 = 7 ✓

2 Split the middle

We write −7x as −x − 6x.

It is the same amount.

2x² − x − 6x + 3 < 0

3 Group in pairs

We put the first two together, and the last two together.

(2x² − x) + (−6x + 3) < 0

From the first pair we take out x.

From the second pair we take out −3.

Both leave the same bracket.

(2x² − x) = x(2x − 1)

(−6x + 3) = −3(2x − 1)

Now both parts have 2x − 1 in them.

x(2x − 1) − 3(2x − 1) < 0

4 Same bracket out

2x − 1 is in both parts, so we take it out once.

What is left, x − 3, goes in the other bracket.

ab + cb = b(a + c)

So we get (2x − 1)(x − 3) < 0.

(2x − 1)(x − 3) < 0

5 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.

2x − 1 = 0 x − 3 = 0

In 2x − 1 = 0, we add 1 on both sides.

In x − 3 = 0, we add 3 on both sides.

2x − 1 + 1 = 0 + 1 x − 3 + 3 = 0 + 3

Now we have 2x = 1 and x = 3.

2x = 1 x = 3

To remove times 2, we divide both sides by 2.

The critical points are 1/2 and 3.

x = 12 x = 3

6 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 1/2 and 3 split the number line into 3 parts.

At 1/2, 2x − 1 is 0.

So the product is 0 there.

At 3, x − 3 is 0.

So the product is 0 there.

Left of 1/2, we test x = 0.

2x − 1 is −1 and x − 3 is −3.

−1

−3

Minus times minus is plus.

Between 1/2 and 3, we test x = 1.

2x − 1 is 1 and x − 3 is −2.

−2

Plus times minus is minus.

Right of 3, we test x = 4.

2x − 1 is 7 and x − 3 is 1.

Plus times plus is plus.

7 Read the answer

Less than 0 means negative.

So we want the parts with a minus.

So 1/2 < x < 3.

12 < x < 3

< 0 → the − parts

Careful!

Not the plus parts!

Less than 0 means minus.

Careful: not x < 12 or x > 3 ✗

8 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: (12, 3)

9 Check

Let's check with a picture: the graph of y = 2x² − 7x + 3.

It is a parabola.

It crosses the x-axis at the critical points, 1/2 and 3.

Below the x-axis, y is minus.

That is 1/2 < x < 3: our answer.

Let's check: a number from the answer, 1, put into the question.

x = 1: 2(1)² − 7(1) + 3 = 2 − 7 + 3 = −2

−2 is less than 0.

True!

−2 < 0 true

Let's look back.

First: find two numbers.

Then: split the middle.

Then: group in pairs.

Then: same bracket out.

Then: find the critical points.

Then: make a sign table.

Then: read the answer.

Then: show it on a number line.

Last: check.

Answer