Solve each equation. Rewrite the equation as two cases.
1). The absolute value of x - 2 minus 3 equals 5.
A: I know one of the solutions is 10. However, I do not believe there is a second solution.
2). The absolute value of x + 7 increased by 2 = 10.
A: I know one of the solutions is 1. However, I am uncertain of what the other is.
1 )
| x - 2 | - 3 = 5 Add 3 to both sides
| x - 2 | - 3 + 3 = 5 + 3
| x - 2 | = 8
x - 2 = + OR - 8
First solution :
x - 2 = 8 Add 2 to both sides
x - 2 + 2 = 8 + 2
x = 10
Second solution :
x - 2 = - 8 Add 2 to both sides
x - 2 + 2 = - 8 + 2
x = - 6
Solutions :
x = - 6 and x = 10
2 )
| x + 7 | + 2 = 10 Subtract 2 to both sides
| x + 7 | + 2 - 2 = 10 - 2
| x + 7 | = 8
x + 7 = + OR - 8
First solution :
x + 7 = 8 Subtract 7 to both sides
x + 7 - 7 = 8 - 7
x = 1
Second solution :
x + 7 = - 8 Subtract 7 to both sides
x + 7 - 7 = - 8 - 7
x = - 15
Solutions :
x = 1 and x = - 15
To solve the equations and rewrite them as two cases, we need to isolate the absolute value expression and solve for both positive and negative cases. Let's break down each equation step by step:
1) The absolute value of x - 2 minus 3 equals 5:
To isolate the absolute value expression, we need to move the constant term outside the absolute value brackets.
| x - 2 | - 3 = 5
Now, we can rewrite this equation as two separate cases, one for the positive and one for the negative:
Case 1 - Positive:
x - 2 - 3 = 5
x - 5 = 5
x = 5 + 5
x = 10
So one of the solutions is x = 10.
Case 2 - Negative:
x - 2 - 3 = -5
x - 5 = -5
x = -5 + 5
x = 0
However, when we substitute this into the original equation, we find that it doesn't satisfy the equation:
|0 - 2| - 3 = 5
|-2| - 3 = 5
2 - 3 = 5
-1 = 5
Since the equation doesn't hold for this case, there is no second solution.
Therefore, the equation x - 2 - 3 = 5 can be rewritten as:
x = 10 (Case 1)
2) The absolute value of x + 7 increased by 2 equals 10:
To isolate the absolute value expression, we need to move the constant term outside the absolute value brackets.
| x + 7 | + 2 = 10
Now, we can rewrite this equation as two separate cases, one for the positive and one for the negative:
Case 1 - Positive:
x + 7 + 2 = 10
x + 9 = 10
x = 10 - 9
x = 1
So one of the solutions is x = 1.
Case 2 - Negative:
-(x + 7) + 2 = 10
-x - 7 + 2 = 10
-x - 5 = 10
-x = 10 + 5
-x = 15
x = -15
However, when we substitute this into the original equation, we find that it doesn't satisfy the equation:
|-15 + 7 | + 2 = 10
|-8| + 2 = 10
8 + 2 = 10
10 = 10
Since the equation holds for this case, the second solution is x = -15.
Therefore, the equation | x + 7 | + 2 = 10 can be rewritten as:
x = 1 (Case 1)
x = -15 (Case 2)