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

Step by step

The question

Question: solve 2x² + x − 3 ≤ 0.

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

2x² + x − 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 1.

2 × (−3) = −6

? × ? = −6, ? + ? = 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 6, smallest first.

We take the smaller from the bigger.

We need 1.

2 and 3 give 1.

Yes!

With the signs: −2 and 3.

1 × 6: 6 − 1 = 5

2 × 3: 3 − 2 = 1 ✓

2 Split the middle

We write x as −2x + 3x.

It is the same amount.

2x² − 2x + 3x − 3 ≤ 0

3 Group in pairs

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

(2x² − 2x) + (3x − 3) ≤ 0

From the first pair we take out 2x.

From the second pair we take out 3.

Both leave the same bracket.

(2x² − 2x) = 2x(x − 1)

(3x − 3) = 3(x − 1)

Now both parts have x − 1 in them.

2x(x − 1) + 3(x − 1) ≤ 0

4 Same bracket out

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

What is left, 2x + 3, goes in the other bracket.

ab + cb = b(a + c)

So we get (x − 1)(2x + 3) ≤ 0.

(x − 1)(2x + 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.

x − 1 = 0 2x + 3 = 0

In x − 1 = 0, we add 1 on both sides.

In 2x + 3 = 0, we take away 3 on both sides.

x − 1 + 1 = 0 + 1 2x + 3 − 3 = 0 − 3

Now we have x = 1 and 2x = −3.

x = 1 2x = −3

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

The critical points are 1 and −3/2.

x = 1 x = −32

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

At −3/2, 2x + 3 is 0.

So the product is 0 there.

At 1, x − 1 is 0.

So the product is 0 there.

Left of −3/2, we test x = −2.

−2

x − 1 is −3 and 2x + 3 is −1.

−3

−1

Minus times minus is plus.

Between −3/2 and 1, we test x = 0.

x − 1 is −1 and 2x + 3 is 3.

−1

Minus times plus is minus.

Right of 1, we test x = 2.

x − 1 is 1 and 2x + 3 is 7.

Plus times plus is plus.

7 Read the answer

Less than or equal to 0 means negative or 0.

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

So −3/2 ≤ x ≤ 1.

−32 ≤ x ≤ 1

≤ 0 → the − parts and the 0s

Careful!

Not the plus parts!

Less than 0 means minus.

Careful: not x ≤ −32 or x ≥ 1 ✗

8 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: [−32, 1]

9 Check

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

It is a parabola.

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

Below the x-axis, y is minus.

That is −3/2 ≤ x ≤ 1: our answer.

Let's check: a number from the answer, 0, put into the question.

x = 0: 2(0)² + 0 − 3 = 0 + 0 − 3 = −3

−3 is less than or equal to 0.

True!

−3 ≤ 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