Show all steps solve the equation check for extraneous solution 9|9-8 x|=2x+3
9|9-8 x|=2x+3 , x ≥ -3/2
|9-8 x|=(2x+3)/9
9-8x = (2x+3)/9
81 - 72x = 2x+3
-74x = -78
x = -78/-74
x = 39/37
or
9-8x = -(2x+3)/9
81 - 72x = -2x - 3
-70x = -84
x = 87/70 = 6/5
check:
if x = 6/5
LS = 9|9-8(6/5)|
= 27/5
RS = 2(6/5)+3
= 27/5 , so x = 6/5
if x = 39/37
LS = 9|9-8(39/37)|
= 189/37
RS = 2(39/37) + 3 = 189/37 , so x = 39/37
so yes?
To solve the equation 9|9-8x|=2x+3 and check for extraneous solutions, follow these steps:
Step 1: Remove the absolute value by separating the equation into two cases, one with a positive argument and one with a negative argument.
Case 1: 9-8x > 0
Case 2: 9-8x < 0
Step 2: Solve the equation for each case separately.
Case 1:
9-8x > 0
-8x > -9
x < 9/8
Case 2:
9-8x < 0
-8x < -9
x > 9/8
Step 3: Check if the solutions obtained from each case satisfy the original equation.
Take the solutions x < 9/8 and x > 9/8 and substitute them into the original equation 9|9-8x|=2x+3.
For x < 9/8:
9|9-8(9/8)|=2(9/8)+3
9|9-9|=18/8+24/8
9(0)=42/8
0=42/8 (Not true)
For x > 9/8:
9|9-8(9/8)|=2(9/8)+3
9|9-9|=18/8+24/8
9(0)=42/8
0=42/8 (Not true)
Step 4: Since neither solution satisfies the original equation, there are no valid solutions. Therefore, the equation has no solution.
In summary, the steps to solve the equation 9|9-8x|=2x+3 and check for extraneous solutions can be broken down as follows:
1. Remove the absolute value by splitting the equation into two cases.
2. Solve each case separately.
3. Substitute the obtained solutions into the original equation and check for validity.
4. Conclude whether there are valid solutions or not. In this case, there are no valid solutions.
To solve the equation 9|9-8x|=2x+3, we will go through the following steps:
Step 1: Remove the absolute value bars by considering both cases (positive and negative) for the expression inside the absolute value.
Case 1: If 9 - 8x > 0, then the equation becomes 9 - 8x = 2x + 3.
Start by subtracting 9 from both sides: -8x = 2x + 3 - 9.
Simplify: -8x = 2x - 6.
Next, subtract 2x from both sides: -10x = -6.
Divide both sides by -10 to solve for x: x = (-6)/(-10) = 3/5.
Case 2: If 9 - 8x < 0, then the equation becomes -(9 - 8x) = 2x + 3.
Start by distributing the negative sign: -9 + 8x = 2x + 3.
Next, subtract 2x from both sides: 6x - 9 = 3.
Then, add 9 to both sides: 6x = 3 + 9.
Simplify: 6x = 12.
Divide both sides by 6 to solve for x: x = 12/6 = 2.
Step 2: Check for extraneous solutions by substituting the obtained solutions back into the original equation.
For x = 3/5:
Original equation: 9|9-8x| = 2x + 3.
Substituting x = 3/5, we get: 9|9 - 8(3/5)| = 2(3/5) + 3.
Simplifying, we have: 9|9 - 24/5| = 6/5 + 3.
Further simplifying gives: 9|45/5 - 24/5| = 6/5 + 15/5.
Continuing to simplify, we obtain: 9|21/5| = 21/5.
Dividing the absolute value term, we have: 9 * (21/5) = 21/5.
Simplifying the left side yields: 189/5 = 21/5.
The equation is true.
For x = 2:
Original equation: 9|9-8x| = 2x + 3.
Substituting x = 2, we get: 9|9 - 8(2)| = 2(2) + 3.
Simplifying, we have: 9|9 - 16| = 4 + 3.
Further simplifying gives: 9|-7| = 7.
Evaluating the absolute value term, we have: 9 * 7 = 7.
Simplifying the left side yields: 63 = 7.
This equation is false.
Step 3: Final solution.
We found two candidate solutions, x = 3/5 and x = 2. However, after checking for extraneous solutions, we see that only x = 3/5 is a valid solution. The extraneous solution x = 2 does not satisfy the original equation.