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