The question
Question: solve x + 3 > 7.
This is an inequality.
The sign means greater than.
x + 3 > 7
1 Remove the + 3
We want x by itself.
The + 3 is in the way.
To remove + 3, we take away 3.
We do it on both sides.
Why both sides?
The sign compares the two sides.
If we change only one side, the answer can change.
x + 3 − 3 > 7 − 3
+ 3 and − 3 make zero.
Cross them out.
7 − 3 = 4.
7 − 3 = 4
So x is greater than 4.
x > 4
2 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.
∞ means it goes on for ever.
interval: (4, ∞)
3 Check
Let's test one number from the answer: 5.
We put it in place of x.
First the left side.
Left: 5 + 3 = 8
Now the right side.
Right: 7
8 is greater than 7.
True!
So the answer works.
8 > 7
Let's look back.
First: remove the + 3.
Then: show it on a number line.
Last: check.
▶ Watch it solved
The question
Question: solve 5 − 2x ≥ 1.
This is an inequality.
The sign means greater than or equal to.
5 − 2x ≥ 1
We want x by itself.
Two things are stuck to x: times −2 and + 5.
The + 5 is on the outside, so it comes off first.
Then the times −2.
1 Remove the 5
To remove 5, we take away 5.
We do it on both sides.
Why both sides?
The sign compares the two sides.
If we change only one side, the answer can change.
5 − 2x − 5 ≥ 1 − 5
5 and − 5 make zero.
Cross them out.
1 − 5 = −4.
1 − 5 = −4
Now we have −2x ≥ −4.
−2x ≥ −4
2 Remove the −2
−2x means −2 times x.
We want just x.
To remove times −2, we divide by −2.
We do it on both sides.
We divide by a negative number.
That turns the sign around.
Greater than or equal to becomes less than or equal to.
−2x−2 ≤ −4−2
−2 divided by −2 is 1.
So just x is left.
−4 divided by −2 is 2.
−4−2 = 2
So x is less than or equal to 2.
x ≤ 2
Careful!
Many people forget to turn the sign.
They get x ≥ 2.
But we divided by a negative number.
Careful: not x ≥ 2 ✗
3 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.
∞ means it goes on for ever.
interval: (−∞, 2]
4 Check
Let's test one number from the answer: 1.
We put it in place of x.
First the left side.
Left: 5 − 2(1) = 5 − 2 = 3
Now the right side.
Right: 1
3 is greater than or equal to 1.
True!
So the answer works.
3 ≥ 1
Let's look back.
First: remove the 5.
Then: remove the −2.
Then: show it on a number line.
Last: check.
Answer Careful: not x ≥ 2 ✗
▶ Watch it solved
The question
Question: solve 3(2 − x) < x + 2.
This is an inequality.
The sign means less than.
3(2 − x) < x + 2
1 Open the brackets
The 3 in front of the bracket multiplies every part inside.
So we write 3 times each part.
3 × 2 + 3 × (−x) < x + 2
Work out each multiplication.
6 − 3x < x + 2
2 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the −3x from the left side.
To remove −3x, we add 3x.
We do it on both sides.
Why both sides?
The sign compares the two sides.
If we change only one side, the answer can change.
6 − 3x + 3x < x + 2 + 3x
− 3x and + 3x make zero.
Cross them out.
x + 3x = 4x.
x + 3x = 4x
Now we have 6 < 4x + 2.
6 < 4x + 2
Let's swap the two sides, so x is on the left.
When we swap, the sign turns around too.
Less than becomes greater than.
4x + 2 > 6
We want x by itself.
Two things are stuck to x: times 4 and + 2.
The + 2 is on the outside, so it comes off first.
Then the times 4.
3 Remove the + 2
To remove + 2, we take away 2.
We do it on both sides, so the sign stays true.
4x + 2 − 2 > 6 − 2
+ 2 and − 2 make zero.
Cross them out.
6 − 2 = 4.
6 − 2 = 4
Now we have 4x > 4.
4x > 4
4 Remove the 4
4x means 4 times x.
We want just x.
To remove times 4, we divide by 4.
We do it on both sides, so the sign stays true.
4x4 > 44
4 divided by 4 is 1.
So just x is left.
4 divided by 4 is 1.
44 = 1
So x is greater than 1.
x > 1
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.
∞ means it goes on for ever.
interval: (1, ∞)
6 Check
Let's test one number from the answer: 2.
We put it in place of x.
First the left side.
Left: 3(2 − 2) = 3(0) = 0
Now the right side.
Right: 2 + 2 = 4
0 is less than 4.
True!
So the answer works.
0 < 4
Let's look back.
First: open the brackets.
Then: get x on one side.
Then: remove the + 2.
Then: remove the 4.
Then: show it on a number line.
Last: check.
▶ Watch it solved
