Year 11 2 unit question: absolute values

i had this question during a test and it didn't seem to work. i have tried to solve it since and still nothing. have i done something wrong or is it the question.

/x-1/=2x-1

try and work it out and let me know what x value you get

You just have to set up two different equations. One in which |x-1| is positive, and one in which it is negative. Here's the first.

|x - 1| = 2x - 1 (we'll keep this first one positive)
(x - 1) = 2x - 1
x - 1 = 2x - 1 (add one)
x = 2x (subtract x)
0 = x

To check if this answer is possible, we plug it back into the original equation.

|0 - 1| = 2(0) - 1
|-1| = -1
1 =/= -1

This equation is not true, so x CAN NOT be 0. Let's try making |x - 1| negative.

|x - 1| = 2x - 1 (add your negative sign)
-(x - 1) = 2x - 1
- x + 1 = 2x - 1 (add x)
1 = 3x - 1 (add 1)
2 = 3x
2/3 = x

Now we'll check this one.

|2/3 - 1| = 2(2/3) - 1 (we'll multiply that two on the right side and make our 1s into 3/3 so we can subtract)
|2/3 - 3/3| = 4/3 - 3/3 (subtract)
|-1/3| = 1/3 (take the absolute value)
1/3 = 1/3

This proves true! So x is, in fact, 2/3.

Let's start with an easy one so I have an understanding of what absolute values are:

|x| = 4

Since we know x can be -4 or +4 to give us the answer of 4, when working absolute value problems, when we remove the absolute value from the equaition we ± the other side.

so x = ±4 in the simple problem.

Now for yours.

|x - 1| = 2x - 1

Same steps as my easy one: Remove the absolute value and set the other half to ± itself:

x - 1 = ± (2x - 1)

So this gives us two equations now:

x - 1 = 2x - 1 and x - 1 = -(2x - 1)

Now solve each equation to get its x:

-x = 0 and x - 1 = -2x + 1
x = 0 and 3x = 2
x = 0 and x = 2/3

Your two solutions to the problem are x = 0 and 2/3