Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
Solve the equation. Check for extraneous solutions.
9|9−8x|=2x+3
To solve the equation, we'll start by removing the absolute value sign.
Case 1: 9 - 8x ≥ 0 (positive value inside the absolute value)
9 - 8x = 2x + 3
Let's solve for x:
9 - 8x = 2x + 3
-8x - 2x = 3 - 9
-10x = -6
x = (-6)/(-10)
x = 3/5
Case 2: 9 - 8x < 0 (negative value inside the absolute value)
-(9 - 8x) = 2x + 3
Let's solve for x:
-(9 - 8x) = 2x + 3
-9 + 8x = 2x + 3
8x - 2x = 3 + 9
6x = 12
x = 12/6
x = 2
Therefore, the solution to the equation is x = 3/5 and x = 2.
To check for extraneous solutions, we substitute both values back into the original equation:
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| = 21/5 + 15/5
9(21/5) = 36/5
189/5 = 36/5
The equation is false, so x = 3/5 is an extraneous solution.
For x = 2:
9|9 - 8(2)| = 2(2) + 3
9|9 - 16| = 4 + 3
9|-7| = 7
9(7) = 7
The equation is true, so x = 2 is a valid solution.
Therefore, the only solution to the equation is x = 2.
To solve the equation 9|9−8x|=2x+3 and check for extraneous solutions, follow these steps:
Step 1: Remove the absolute value bars by considering two cases:
Case 1: 9−8x ≥ 0
In this case, the absolute value is equal to the expression inside.
9 - 8x = 2x + 3
Simplify the equation by combining like terms:
-8x - 2x = 3 - 9
-10x = -6
Divide both sides by -10 to solve for x:
x = (-6) / (-10)
Simplify the fraction:
x = 3/5
Check: Substitute the found value of x back into the original equation:
9|9−8x| = 2x + 3
9|9−8(3/5)| = 2(3/5) + 3
Evaluate the absolute value:
9|9 − 24/5| = 6/5 + 3
Simplify the expression inside the absolute value:
9|45/5 − 24/5| = 6/5 + 3
9|21/5| = 6/5 + 3
Evaluate the absolute value:
9(21/5) = 6/5 + 3
Multiply:
189/5 = 6/5 + 3
Common denominator:
189/5 = 6/5 + 15/5
Add the fractions:
189/5 = 21/5
The equation is true, so x = 3/5 is a valid solution for case 1.
Case 2: 9−8x < 0
In this case, the absolute value is the negation of the expression inside.
-(9 - 8x) = 2x + 3
Distribute the negative sign:
-9 + 8x = 2x + 3
Simplify the equation by combining like terms:
8x - 2x = 3 + 9
6x = 12
Divide both sides by 6 to solve for x:
x = 12 / 6
Simplify the fraction:
x = 2
Check: Substitute the found value of x back into the original equation:
9|9−8x| = 2x + 3
9|9−8(2)| = 2(2) + 3
Evaluate the absolute value:
9|9 − 16| = 4 + 3
Simplify the expression inside the absolute value:
9|-7| = 4 + 3
Evaluate the absolute value:
9(7) = 4 + 3
Multiply:
63 = 4 + 3
Sum:
63 = 7
The equation is false, so x = 2 is an extraneous solution for case 2.
Step 2: Final Solution
The solutions to the equation 9|9−8x|=2x+3 are x = 3/5 and x = 2.
However, x = 2 is an extraneous solution because it does not satisfy the original equation. Therefore, the only valid solution is x = 3/5.
To solve the equation 9|9−8x|=2x+3, we need to isolate the absolute value term and then solve for x.
Step 1: Eliminate the absolute value by considering the positive and negative cases.
Case 1: 9−8x ≥ 0 (when the expression inside the absolute value is positive)
In this case, the absolute value can be removed, so the equation becomes:
9−8x = 2x+3
Case 2: 9−8x < 0 (when the expression inside the absolute value is negative)
When the expression inside the absolute value is negative, we need to remove the absolute value and multiply that side of the equation by -1. Therefore, the equation becomes:
-(9−8x) = 2x+3
Step 2: Solve for x in both cases.
Case 1: 9−8x = 2x+3
Combining like terms, we get:
-8x - 2x = 3 - 9
-10x = -6
Dividing both sides by -10, we have:
x = -6/-10
x = 3/5
Case 2: -(9−8x) = 2x+3
Distribute the negative sign inside the parentheses, and we have:
-9+8x = 2x+3
Combine like terms, and we get:
8x - 2x = 3 + 9
6x = 12
Dividing both sides by 6, we have:
x = 12/6
x = 2
Step 3: Check for extraneous solutions.
To check for extraneous solutions, substitute both x = 3/5 and x = 2 back into the original equation and see if they satisfy the equation.
Checking x = 3/5:
9|9−8(3/5)| = 2(3/5)+3
9|9−24/5| = 6/5+3
Simplifying, we get:
9|45/5−24/5| = 6/5+15/5
9|21/5| = 21/5
9(21/5) = 21/5
189/5 = 21/5
The equation is true when x = 3/5.
Checking x = 2:
9|9−8(2)| = 2(2)+3
9|9−16| = 4+3
Simplifying, we get:
9|-7| = 7
63 = 7
The equation is not true when x = 2.
Step 4: Conclusion
The equation 9|9−8x|=2x+3 has one solution, x = 3/5. The solution x = 2 is extraneous and does not satisfy the equation.