Enter your answer and show all the steps that you use to solve this problem in the space provided. Solve the equation. 2|3x+5|=10x+6
To solve the equation, we will need to remove the absolute value bars and solve for x.
1. Start by isolating the absolute value expression by dividing both sides of the equation by 2:
|3x + 5| = (10x + 6)/2
2. Simplify the right side:
|3x + 5| = 5x + 3
3. Now, consider two cases:
a) When 3x + 5 is positive:
In this case, the absolute value expression can be rewritten without the absolute value bars:
3x + 5 = 5x + 3
b) When 3x + 5 is negative:
In this case, the absolute value expression can be rewritten with the opposite sign:
-(3x + 5) = 5x + 3
4. Solve each case separately:
For case a:
3x + 5 = 5x + 3
Rearrange the equation:
3x - 5x = 3 - 5
-2x = -2
Divide both sides by -2:
x = -2/-2
x = 1
For case b:
-(3x + 5) = 5x + 3
Distribute the negative sign:
-3x - 5 = 5x + 3
Rearrange the equation:
-3x - 5x = 3 + 5
-8x = 8
Divide both sides by -8:
x = 8/-8
x = -1
5. Check the solutions:
Substitute each solution into the original equation to verify if they are valid.
For x = 1:
2|3(1) + 5| = 10(1) + 6
2|3 + 5| = 10 + 6
2|8| = 16
2(8) = 16
16 = 16
The solution x = 1 satisfies the equation.
For x = -1:
2|3(-1) + 5| = 10(-1) + 6
2| -3 + 5| = -10 + 6
2| 2| = -4
2(2) = -4
4 = -4
The solution x = -1 does not satisfy the equation.
6. Conclusion:
The only solution to the equation 2|3x+5| = 10x+6 is x = 1.