Solve each absolute value equation.
|x - 3| = 1
Okay here is what I did:
x - 3 = 1
x = 4 or x = -4
The book says {2,4}
I'm very lost. Can someone help me?
|x-3| = 1
Because of the absolute value, x-3 can equal 1 or - 1.
x-3 = 1 or x-3 = -1
x = ?
Ah I see, but I thought that an absolute value cannot equal a negative number. Maybe I'm just over thinking it.
To eliminate the absolute value you must consider the possibilities that the contents of the absolute value were positive or negative.
The absolute value cannot equal a negative - you will notice that while |x-3| = 1, x-3 = 1 or -1
Sure, I can help you understand how to solve the absolute value equation |x - 3| = 1. Let me break it down for you step by step.
Step 1: Setting up two equations
When you have an absolute value equation, there are always two possible cases: one where the expression inside the absolute value is positive, and another where it is negative. So, we write two equations to cover both cases:
1) x - 3 = 1, if (x - 3) is positive.
2) -(x - 3) = 1, if (x - 3) is negative.
Step 2: Solve the first equation
To solve the equation x - 3 = 1, you can add 3 to both sides of the equation to isolate x:
x - 3 + 3 = 1 + 3
x = 4
Step 3: Solve the second equation
For the equation -(x - 3) = 1, you need to distribute the negative sign:
-x + 3 = 1
Now, subtract 3 from both sides of the equation:
-x + 3 - 3 = 1 - 3
-x = -2
To isolate x, multiply both sides of the equation by -1:
-x * (-1) = -2 * (-1)
x = 2
Step 4: Check Solutions
Now, we have two possible solutions, x = 4 and x = 2. To verify if these solutions are correct, substitute them back into the original equation |x - 3| = 1.
For x = 4:
|4 - 3| = 1
|1| = 1
1 = 1 (True)
For x = 2:
|2 - 3| = 1
|-1| = 1
1 = 1 (True)
Both solutions satisfy the original equation, so the correct solution set is {2, 4}.