Solve the equation. Check for extraneous solutions.
9|9 - 8x| = 2x + 3
Let's solve the equation step by step:
1. Remove the absolute value by splitting it into two separate cases:
9 - 8x = 2x + 3 or 9 - 8x = -(2x + 3)
2. Solve the first case:
9 - 8x = 2x + 3
Move all terms with x to one side:
9 - 3 = 2x + 8x
Combine like terms:
6 = 10x
Divide by 10:
6/10 = x
Simplify: 3/5 = x
3. Solve the second case:
9 - 8x = -(2x + 3)
Remove the negative sign by multiplying both sides by -1:
-(9 - 8x) = -(-2x - 3)
Distribute the negative sign:
-9 + 8x = 2x + 3
Move all terms with x to one side:
-9 - 3 = 2x - 8x
Combine like terms:
-12 = -6x
Divide by -6:
-12/-6 = x
Simplify: 2 = x
Checking for extraneous solutions:
Plug x = 3/5 into the original equation:
9|9 - 8(3/5)| = 2(3/5) + 3
Simplify:
9|9 - 24/5| = 6/5 + 3
Evaluate the absolute value:
9|45/5 - 24/5| = 6/5 + 3
Simplify:
9|21/5| = 6/5 + 15/5
Evaluate the absolute value:
9(21/5) = 21/5 + 15/5
Simplify:
189/5 = 36/5
Since the equation is not true, x = 3/5 is not a valid solution.
Plug x = 2 into the original equation:
9|9 - 8(2)| = 2(2) + 3
Simplify:
9|9 - 16| = 4 + 3
Evaluate the absolute value:
9|-7| = 4 + 3
Simplify:
9(7) = 7
Since the equation is true, x = 2 is a valid solution.
Therefore, the solution to the equation is x = 2.
To solve the equation, we need to isolate the absolute value expression first. Let's break down the equation into two cases: one where the expression inside the absolute value is positive and the other where it is negative.
Case 1: 9 - 8x ≥ 0
If 9 - 8x ≥ 0, then we can remove the absolute value signs. Rewrite the equation as:
9 - 8x = 2x + 3
Now, let's solve for x:
Subtract 3 from both sides:
9 - 8x - 3 = 2x + 3 - 3
6 - 8x = 2x
Add 8x to both sides:
6 - 8x + 8x = 2x + 8x
6 = 10x
Divide by 10:
6/10 = 10x/10
3/5 = x
So, in case 1, we have x = 3/5.
Case 2: 9 - 8x < 0
If 9 - 8x < 0, then the expression inside the absolute value becomes negated. Rewrite the equation as:
-(9 - 8x) = 2x + 3
Distribute the negative sign to the terms inside the parentheses:
-9 + 8x = 2x + 3
Now, let's solve for x:
Subtract 2x from both sides:
-9 + 8x - 2x = 2x - 2x + 3
-9 + 6x = 3
Add 9 to both sides:
-9 + 6x + 9 = 3 + 9
6x = 12
Divide by 6:
6x/6 = 12/6
x = 2
So, in case 2, we have x = 2.
Now, let's check for extraneous solutions. We substitute the values of x back into the original equation and see if it holds true.
For x = 3/5:
9|9 - 8(3/5)| = 2(3/5) + 3
9|9 - 24/5| = 6/5 + 3
9|45/5 - 24/5| = 6/5 + 3
9|21/5| = 6/5 + 15/5
9 * 21/5 = 21/5 + 15/5
189/5 = 36/5
The equation holds true for x = 3/5.
For x = 2:
9|9 - 8(2)| = 2(2) + 3
9|9 - 16| = 4 + 3
9|-7| = 7
9(7) = 7
The equation also holds true for x = 2.
Therefore, the solutions to the equation are x = 3/5 and x = 2.