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