Step by step
The question
Question: solve −4t + 3 > 15.
This is an inequality.
The sign means greater than.
We want t by itself.
Two things are stuck to t: times −4 and + 3.
The + 3 is on the outside, so it comes off first.
Then the times −4.
1 Remove the + 3
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.
+ 3 and − 3 make zero.
Cross them out.
15 − 3 = 12.
Now we have −4t > 12.
2 Remove the −4
−4t means −4 times t.
We want just t.
To remove times −4, we divide by −4.
We do it on both sides.
We divide by a negative number.
That turns the sign around.
Greater than becomes less than.
−4 divided by −4 is 1.
So just t is left.
12 divided by −4 is −3.
So t is less than −3.
Careful!
Many people forget to turn the sign.
They get t > −3.
But we divided by a negative number.
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.
∞ means it goes on for ever.
4 Check
Let's test one number from the answer: −4.
We put it in place of t.
First the left side.
Now the right side.
19 is greater than 15.
True!
So the answer works.
Let's look back.
First: remove the + 3.
Then: remove the −4.
Then: show it on a number line.
Last: check.