Solve the equation. Check for extraneous solutions. 9|9−8x|=2x+3
We need to solve the equation 9|9-8x|=2x+3.
First, let's remove the absolute value by breaking the equation into two cases:
Case 1: 9-8x ≥ 0
If 9-8x ≥ 0, then we can simplify the equation to 9(9-8x) = 2x+3.
Expanding the left side of the equation, we get 81-72x = 2x+3.
Combining like terms, we have -72x-2x = 3-81.
Simplifying further, we get -74x = -78.
Dividing both sides of the equation by -74, we obtain x = 78/74.
Simplifying the fraction, we find x = 39/37.
Checking for extraneous solutions:
Substituting x = 39/37 back into the original equation, we have 9|9-8(39/37)| = 2(39/37)+3.
Simplifying, we get 9|9-312/37| = 78/37+3.
Simplifying further, we have 9|333/37-312/37| = 189/37.
Finding the absolute value, we get 9|(333-312)/37| = 189/37.
Simplifying, we have 9|21/37| = 189/37.
Since |21/37| = 21/37, we have 9(21/37) = 189/37.
Simplifying, we get 189/37 = 189/37.
The equation is true, so x = 39/37 is a valid solution.
Case 2: 9-8x < 0
If 9-8x < 0, then we can simplify the equation to 9(-9+8x) = 2x+3.
Expanding the left side of the equation, we get -81+72x = 2x+3.
Combining like terms, we have 72x-2x = 3+81.
Simplifying further, we get 70x = 84.
Dividing both sides of the equation by 70, we obtain x = 84/70.
Simplifying the fraction, we find x = 12/10.
Reducing the fraction, we have x = 6/5.
Checking for extraneous solutions:
Substituting x = 6/5 back into the original equation, we have 9|9-8(6/5)| = 2(6/5)+3.
Simplifying, we get 9|9-48/5| = 12/5+3.
Simplifying further, we have 9|45/5-48/5| = 27/5.
Finding the absolute value, we get 9|-3/5| = 27/5.
Since |-3/5| = 3/5, we have 9(3/5) = 27/5.
Simplifying, we get 27/5 = 27/5.
The equation is true, so x = 6/5 is a valid solution.
Therefore, the equation 9|9-8x|=2x+3 has two solutions: x = 39/37 and x = 6/5.
show the steps more simply
Sure! Here are the steps in a simpler form:
1. Start with the equation 9|9-8x|=2x+3.
2. Break the equation into two cases: 9-8x ≥ 0 and 9-8x < 0.
3. Solve the first case, 9-8x ≥ 0:
- Simplify the equation to 9(9-8x) = 2x+3.
- Expand the left side: 81-72x = 2x+3.
- Combine like terms: -72x-2x = 3-81.
- Simplify further: -74x = -78.
- Divide both sides by -74: x = 78/74 or x = 39/37.
- Check the extraneous solution: Substitute x = 39/37 back into the original equation and verify it satisfies the equation.
4. Solve the second case, 9-8x < 0:
- Simplify the equation to 9(-9+8x) = 2x+3.
- Expand the left side: -81+72x = 2x+3.
- Combine like terms: 72x-2x = 3+81.
- Simplify further: 70x = 84.
- Divide both sides by 70: x = 84/70 or x = 12/10 or x = 6/5.
- Check the extraneous solution: Substitute x = 6/5 back into the original equation and verify it satisfies the equation.
5. Therefore, the equation 9|9-8x|=2x+3 has two solutions: x = 39/37 and x = 6/5.
show the steps without explanation
Sure! Here are the steps without explanation:
1. Start with the equation 9|9-8x|=2x+3.
2. Break the equation into two cases: 9-8x ≥ 0 and 9-8x < 0.
3. Solve the first case, 9-8x ≥ 0: x = 39/37
4. Solve the second case, 9-8x < 0: x = 6/5
5. Therefore, the equation 9|9-8x|=2x+3 has two solutions: x = 39/37 and x = 6/5.