Worked examples
The question
Question: solve 1 < x + 4 < 6.
x is in the middle, between two numbers.
1 < x + 4 < 6
1 Remove the + 4
We want x by itself.
The + 4 is in the way.
To remove + 4, we take away 4.
We do it to all three parts.
Why all three?
The signs compare the three parts.
If we change only one part, the answer can change.
1 − 4 < x + 4 − 4 < 6 − 4
+ 4 and − 4 make zero.
Cross them out.
1 minus 4 is −3.
6 minus 4 is 2.
1 − 4 = −3
6 − 4 = 2
Now we have −3 < x < 2.
So x is between −3 and 2: −3 < x < 2.
−3 < x < 2
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.
interval: (−3, 2)
3 Check
Let's test one number from the answer: 1.
We put it in place of x in the middle part.
Middle: 1 + 4 = 5
5 is greater than 1, and less than 6.
True!
So the answer works.
1 < 5 < 6
Let's look back.
First: remove the + 4.
Then: show it on a number line.
Last: check.
▶ Watch it solved
The question
Question: solve −3 < 2x + 1 ≤ 7.
x is in the middle, between two numbers.
−3 < 2x + 1 ≤ 7
1 Remove the + 1
We want x by itself.
The + 1 is in the way.
To remove + 1, we take away 1.
We do it to all three parts.
Why all three?
The signs compare the three parts.
If we change only one part, the answer can change.
−3 − 1 < 2x + 1 − 1 ≤ 7 − 1
+ 1 and − 1 make zero.
Cross them out.
−3 minus 1 is −4.
7 minus 1 is 6.
−3 − 1 = −4
7 − 1 = 6
Now we have −4 < 2x ≤ 6.
−4 < 2x ≤ 6
2 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it to all three parts.
2 is a positive number, so the signs stay the same.
−42 < 2x2 ≤ 62
2 divided by 2 is 1.
So just x is left.
−4 divided by 2 is −2.
6 divided by 2 is 3.
−42 = −2, 62 = 3
Now we have −2 < x ≤ 3.
So x is between −2 and 3: −2 < x ≤ 3.
−2 < x ≤ 3
3 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.
A square bracket means the end is in, like the filled circle.
interval: (−2, 3]
4 Check
Let's test one number from the answer: 1.
We put it in place of x in the middle part.
Middle: 2(1) + 1 = 2 + 1 = 3
3 is greater than −3, and less than or equal to 7.
True!
So the answer works.
−3 < 3 ≤ 7
Let's look back.
First: remove the + 1.
Then: remove the 2.
Then: show it on a number line.
Last: check.
▶ Watch it solved
The question
Question: solve −4 ≤ 1 − 5x < 11.
x is in the middle, between two numbers.
−4 ≤ 1 − 5x < 11
1 Remove the 1
We want x by itself.
The 1 is in the way.
To remove 1, we take away 1.
We do it to all three parts.
Why all three?
The signs compare the three parts.
If we change only one part, the answer can change.
−4 − 1 ≤ 1 − 5x − 1 < 11 − 1
1 and − 1 make zero.
Cross them out.
−4 minus 1 is −5.
11 minus 1 is 10.
−4 − 1 = −5
11 − 1 = 10
Now we have −5 ≤ −5x < 10.
−5 ≤ −5x < 10
2 Remove the −5
−5x means −5 times x.
We want just x.
To remove times −5, we divide by −5.
We do it to all three parts.
We divide by a negative number.
That turns both signs around.
−5−5 ≥ −5x−5 > 10−5
−5 divided by −5 is 1.
So just x is left.
−5 divided by −5 is 1.
10 divided by −5 is −2.
−5−5 = 1, 10−5 = −2
Now we have 1 ≥ x > −2.
1 ≥ x > −2
We turn it around, so the small number is on the left.
It says the same thing.
So x is between −2 and 1: −2 < x ≤ 1.
−2 < x ≤ 1
3 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.
A square bracket means the end is in, like the filled circle.
interval: (−2, 1]
4 Check
Let's test one number from the answer: 1.
We put it in place of x in the middle part.
Middle: 1 − 5(1) = 1 − 5 = −4
−4 is greater than or equal to −4, and less than 11.
True!
So the answer works.
−4 ≤ −4 < 11
Let's look back.
First: remove the 1.
Then: remove the −5.
Then: show it on a number line.
Last: check.
▶ Watch it solved
