Solve the equation. Check for extraneous solutions. 9|9-8x|=2x+3
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.
To solve the equation, we need to isolate the variable x. Let's break it down step-by-step.
Step 1: Start by removing the absolute value bars. This will give us two separate equations, one for when the expression inside the absolute value is positive, and one for when it is negative.
For 9 - 8x ≥ 0 (positive case):
When 9 - 8x ≥ 0, we have:
9 - 8x = 2x + 3
Simplifying this equation, we get:
9 - 3 = 2x + 8x
6 = 10x
Dividing both sides by 10, we find:
x = 6/10
x = 3/5
For 9 - 8x < 0 (negative case):
When 9 - 8x < 0, we have:
-(9 - 8x) = 2x + 3
Simplifying this equation, we get:
-9 + 8x = 2x + 3
Subtracting 2x from both sides, we have:
8x - 2x = 3 + 9
6x = 12
Dividing both sides by 6, we find:
x = 12/6
x = 2
Step 2: Now we have two possible solutions, x = 3/5 and x = 2. However, before we finalize it, we need to check for extraneous solutions.
To do that, we substitute each possible solution back into the original equation and check if both sides are equal.
Checking for x = 3/5:
9|(9 - 8(3/5))| = 2(3/5) + 3
9|(9 - 24/5)| = 6/5 + 3
Simplifying both sides:
9|(45/5 - 24/5)| = 6/5 + 15/5
9|(21/5)| = 21/5
Simplifying further:
9 * (21/5) = 21/5
(189/5) = 21/5
The equation holds true for x = 3/5.
Now checking for x = 2:
9|(9 - 8(2))| = 2(2) + 3
9|(9 - 16)| = 4 + 3
Simplifying both sides:
9|-7| = 7
This is not true since the left side is not equal to the right side.
Step 3: In conclusion, the solution to the equation 9|9 - 8x| = 2x + 3 is x = 3/5. The solution x = 2 is an extraneous solution and is not valid.
To solve the equation 9|9-8x|=2x+3, we need to isolate the absolute value expression and break it into two cases: one where the expression inside the absolute value is positive and one where it is negative.
Case 1: (9-8x) is positive: In this case, the absolute value can be eliminated, and the equation becomes:
9 - 8x = 2x + 3
Now, let's solve for x:
-8x - 2x = 3 - 9
-10x = -6
Divide both sides by -10:
x = -6 / -10
x = 3/5
Case 2: (9-8x) is negative: In this case, the expression inside the absolute value becomes its opposite, and the equation becomes:
-(9 - 8x) = 2x + 3
Distribute the negative sign:
-9 + 8x = 2x + 3
Let's solve for x:
8x - 2x = 3 + 9
6x = 12
Divide both sides by 6:
x = 12 / 6
x = 2
So, we have two potential solutions: x = 3/5 and x = 2.
To check if these solutions are extraneous, we substitute them back into the original equation and see if the equation holds true for each value.
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 + 15/5
9|21/5| = 21/5
9 * (21/5) = 21/5
189/5 = 21/5 (True)
For x = 2:
9|9 - 8(2)| = 2(2) + 3
9|9 - 16| = 4 + 3
9|-7| = 7
9 * 7 = 7 (False)
After checking both solutions, we find that only x = 3/5 is a valid solution to the given equation.