Solve the equation. Check for extraneous solutions. 9|9-8x|=2x+3

Answer 1

Let's solve the equation step by step:

1) We start by removing the absolute value. We can do this by getting rid of the absolute value brackets and considering two cases: one where the expression inside the absolute value is positive and another where it is negative.

When 9-8x is positive, |9-8x| simplifies to 9-8x. The equation becomes:

9(9-8x) = 2x + 3

When 9-8x is negative, |9-8x| simplifies to -(9-8x) = -9 + 8x. The equation becomes:

9(-9+8x) = 2x + 3

2) Solving the first case:

9(9-8x) = 2x + 3

81 - 72x = 2x + 3

Combine like terms:

-72x - 2x = 3 - 81

-74x = -78

Divide both sides by -74:

x = -78 / -74

Simplify:

x = 39/37

3) Solving the second case:

9(-9+8x) = 2x + 3

-81 + 72x = 2x + 3

Combine like terms:

72x - 2x = 3 + 81

70x = 84

Divide both sides by 70:

x = 84 / 70

Simplify:

x = 6/5

4) Checking for extraneous solutions:

Let's substitute x = 39/37 and x = 6/5 back into the original equation and see if they satisfy the equation:

Case 1: x = 39/37

9|9-8(39/37)|=2(39/37)+3

9|9-312/37|=78/37+3

9|-279/37 + 312/37|=78/37+111/37

9|33/37|=189/37

9(33/37)=300/37

99/37=300/37

This statement is false, so x = 39/37 is not a solution.

Case 2: x = 6/5

9|9-8(6/5)|=2(6/5)+3

9|9-48/5|=12/5+3

9|-3/5 + 48/5|=12/5+15/5

9|45/5|=27/5

9(9)=135/5

81=135/5

This statement is also false, so x = 6/5 is not a solution.

Therefore, the equation has no valid solutions.