-(x - 4) = -x + k has infinitely many solutions. Find k.

Step by step

The question

Question: for which k does −(x − 4) = −x + k have infinitely many solutions?

Infinitely many solutions means every number works for x.

1 Open the brackets

A minus in front of a bracket changes the sign of every part inside.

2 Compare the x parts

The x parts: x on the left and x on the right.

3 Take away the x part

We add x on both sides.

They stay equal.

−x and + x make zero.

Cross them out.

The x parts are gone.

Now we have 4 = k.

4 Choose k

When the x parts are gone, what is left is either always true or never true.

We want every x to work.

So 4 = k must be always true.

4 = k is true only when k is 4.

So k = 4!

5 Check

Let's check.

We put k = 4 back in.

Try one number, x = 1.

The left side: −(1 − 4) = −1 × (−3) = 3.

The right side: −1 + 4 = 3.

Both sides are 3.

Any other x works the same way: both sides are −x + 4.

So every 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.