Step by step
The question
Question: for which k does 3(2x + 1) = kx + 5 have no solution?
We find the number k that makes it happen.
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 2x is 6x.
3 times 1 is 3.
2 Compare the x parts
The x parts: 6x on the left and kx on the right.
If the x parts were different, taking one away would still leave some x.
Then we could find x: just one x would work.
We want no x to work.
So the x parts must be the same: kx = 6x.
k times x is 6 times x when k is 6.
So k = 6.
Careful!
With k not 6, the x parts are different.
Then just one x works.
We want none.
Now the x parts are the same.
We still have to look at the numbers.
We put k = 6 in.
3 Take away the x part
We take away 6x on both sides.
They stay equal.
6x and − 6x cancel each other.
They make zero.
The x parts are gone.
Now we have 3 = 5.
When the x parts are gone, what is left is either always true or never true.
3 = 5 is never true.
So no x works.
That is what we want.
So k = 6.
4 Check
Let's check.
We put k = 6 back in.
Try one number, x = 1.
The left side: 3(2 × 1 + 1) = 3 × 3 = 9.
The right side: 6 × 1 + 5 = 6 + 5 = 11.
9 and 11 are not equal.
With the same x parts, the left side is always 2 less.
So no x works.
It works!
5 Type it in
The answer box takes 5 characters.
6 has 1.
It fits.
We type 6.
Let's look back.
First: open the brackets.
Then: compare the x parts.
Then: take away the x part.
Then: check.
Last: type it in.