Solve the equation. Check for extraneous solutions. 9|9-8x|=2x+3
simply answer the question
To solve this equation, we'll first solve for the positive value under the absolute value sign and then solve for the negative value.
For the positive value:
9 - 8x = 2x + 3
To isolate the variable, we'll first subtract 2x from both sides and then add 8x to both sides:
9 - 3 = 2x + 8x
6 = 10x
Now divide both sides by 10:
6/10 = x
0.6 = x
For the negative value:
9 - 8x = -(2x + 3)
To eliminate the negative sign, distribute the negative sign to both terms inside the parentheses:
9 - 8x = -2x - 3
Adding 8x to both sides:
9 = 6x - 3
Next, add 3 to both sides:
12 = 6x
Dividing both sides by 6:
12/6 = x
2 = x
Therefore, the solutions to the equation are x = 0.6 and x = 2.
To check for extraneous solutions, we substitute each found value back into the original equation and ensure it satisfies the equation.
For x = 0.6:
9|9 - 8(0.6)| = 2(0.6) + 3
9|9 - 4.8| = 1.2 + 3
9|4.2| = 1.2 + 3
37.8 = 4.2 + 3
37.8 = 7.2
This is not true, so x = 0.6 is an extraneous solution.
For x = 2:
9|9 - 8(2)| = 2(2) + 3
9|9 - 16| = 4 + 3
9|-7| = 4 + 3
9 * 7 = 4 + 3
63 = 7
This is not true, so x = 2 is also an extraneous solution.
Therefore, there are no valid solutions to the equation.
To solve the equation 9|9-8x|=2x+3, we will break it down into two cases and solve each separately.
Case 1: 9-8x ≥ 0 (positive inside the absolute value)
Since 9-8x is positive, we can remove the absolute value signs and rewrite the equation:
9-8x = 2x + 3
First, we will simplify the equation by combining like terms:
-8x - 2x = 3 - 9
-10x = -6
Next, divide both sides of the equation by -10:
x = -6 / -10
x = 0.6
Case 2: 9-8x < 0 (negative inside the absolute value)
In this case, we need to change the sign inside the absolute value and solve for x:
-(9-8x) = 2x + 3
Start by distributing the negative sign:
-9 + 8x = 2x + 3
Combine like terms:
8x - 2x = 3 + 9
6x = 12
Divide both sides of the equation by 6:
x = 12 / 6
x = 2
Now let's check both solutions for extraneous solutions by plugging them back into the original equation:
For x = 0.6:
9|9-8(0.6)| = 2(0.6) + 3
9|9-4.8| = 1.2 + 3
9|4.2| = 1.2 + 3
9(4.2) = 4.2 + 3
37.8 = 7.2, which is not true
For x = 2:
9|9-8(2)| = 2(2) + 3
9|9-16| = 4 + 3
9|-7| = 7
9(7) = 7
63 = 7, which is not true
Therefore, there are no valid solutions to the equation.
To solve the equation 9|9-8x|=2x+3 and check for extraneous solutions, follow these steps:
Step 1: Remove the absolute value by considering two cases:
Case 1: 9 - 8x ≥ 0 (non-negative inside the absolute value)
Case 2: 9 - 8x < 0 (negative inside the absolute value)
Step 2: Solve each case separately and check for extraneous solutions.
Case 1: 9 - 8x ≥ 0
Solve: 9 - 8x = 2x + 3
Subtract 3 from both sides: 6 - 8x = 2x
Add 8x to both sides: 6 = 10x
Divide both sides by 10: x = 6/10 = 3/5
Check for extraneous solutions:
Substitute x = 3/5 into the original equation:
9|9-8(3/5)| = 2(3/5) + 3
Simplify: 9|9 - 24/5| = 6/5 + 3
Calculate: 9|45/5 - 24/5| = 6/5 + 3
Simplify: 9|21/5| = 6/5 + 15/5
Calculate: 9(21/5) = 21/5 + 15/5
Simplify: 189/5 = 36/5
This is not true, so the solution x = 3/5 is extraneous.
Case 2: 9 - 8x < 0
Solve: -(9 - 8x) = 2x + 3
Distribute the negative sign: -9 + 8x = 2x + 3
Subtract 2x from both sides: -9 + 8x - 2x = 2x - 2x + 3
Simplify: 6x - 9 = 3
Add 9 to both sides: 6x = 12
Divide both sides by 6: x = 12/6 = 2
Check for extraneous solutions:
Substitute x = 2 into the original equation:
9|9-8(2)| = 2(2) + 3
Simplify: 9|9 - 16| = 4 + 3
Calculate: 9|-7| = 7
Simplify: 9(7) = 7
This is true, so the solution x = 2 is valid.
Therefore, the solution to the equation 9|9-8x|=2x+3 with checking for extraneous solutions is x = 2.