Step by step
The question
Question: for which k does 3(x + 2) = 3x + k have no solution?
No solution means no number works for x.
1 Open the brackets
The 3 in front of the bracket multiplies every part inside.
So we write 3 times each part.
3 times x is 3x.
3 times 2 is 6.
Careful!
The 3 multiplies every part, the 2 too.
3 times 2 is 6, not 2.
2 Compare the x parts
The x parts: 3x on the left and 3x on the right.
3 Take away the x part
We take away 3x on both sides.
They stay equal.
3x and − 3x make zero.
Cross them out.
The x parts are gone.
Now we have 6 = k.
4 Choose k
When the x parts are gone, what is left is either always true or never true.
We want no x to work.
So 6 = k must be never true.
6 = k is false for every k except 6.
So k can be any number except 6.
5 Check
Let's check with one k that is not 6.
We pick k = 7.
We take away 3x on both sides: 6 = 7.
Never true, so no x works.
It works!
Let's look back.
First: open the brackets.
Then: compare the x parts.
Then: take away the x part.
Then: choose k.
Last: check.