Solve x + 1 >= 3 and 2x - 1 <= 3

Step by step

The question

Question: solve x + 1 ≥ 3 and 2x − 1 ≤ 3.

"And" means both parts must be true.

1 Solve the first part

First part: x + 1 ≥ 3.

We want x by itself.

The + 1 is in the way.

To remove + 1, we take away 1.

We do it on both sides, so the sign stays true.

+ 1 and − 1 make zero.

Cross them out.

3 − 1 = 2.

So x is greater than or equal to 2.

2 Solve the second part

Second part: 2x − 1 ≤ 3.

We want x by itself.

Two things are stuck to x: times 2 and − 1.

The − 1 is on the outside, so it comes off first.

Then the times 2.

To remove − 1, we add 1.

We do it on both sides, so the sign stays true.

− 1 and + 1 make zero.

Cross them out.

3 + 1 = 4.

Now we have 2x ≤ 4.

2x means 2 times x.

We want just x.

To remove times 2, we divide by 2.

We do it on both sides, so the sign stays true.

2 divided by 2 is 1.

So just x is left.

4 divided by 2 is 2.

So x is less than or equal to 2.

3 Show each part on a number line

4 Find the numbers in both

Only one number is shaded on both lines: 2.

So x = 2.

Careful!

A number that is shaded on only one line is not an answer.

It must be on both.

5 Check

Let's test one number from the answer: 2.

"And": it must work in both parts.

First part.

3 is greater than or equal to 3.

True!

Now the second part.

3 is less than or equal to 3.

True!

So the answer works.

Let's look back.

First: solve the first part.

Then: solve the second part.

Then: show each part on a number line.

Then: find the numbers in both.

Last: check.