The absolute value of (three x plus five) equals twelve x plus 8.
What is being asked in the problem and what does that mean? What plan am I going to try?
Then, solve and explain what your answer means.
The problem is asking us to solve for the value of x in the equation involving the absolute value. To solve the equation, we need to find the value of x that satisfies the equation, making both sides of the equation equal.
To solve the equation, we will first isolate the absolute value expression on one side of the equation. We can then split the equation into two cases, one where the expression inside the absolute value is positive, and the other where it is negative.
In the case where the expression inside the absolute value is positive, we can remove the absolute value symbols and solve for x. In the case where the expression inside the absolute value is negative, we need to change the sign inside the absolute value and then solve for x.
Let's solve the equation step by step:
|3x + 5| = 12x + 8
Case 1: 3x + 5 > 0
In this case, we can remove the absolute value symbols without changing the equation: 3x + 5 = 12x + 8.
Simplifying the equation:
3x - 12x = 8 - 5
-9x = 3
x = -3/9
x = -1/3
Case 2: 3x + 5 < 0
In this case, we need to change the sign inside the absolute value: -(3x + 5) = 12x + 8.
Simplifying the equation:
-3x - 5 = 12x + 8
Moving all the terms to one side:
-3x - 12x = 8 + 5
-15x = 13
x = 13/-15
x = -13/15
Therefore, the two solutions for x are x = -1/3 and x = -13/15.
To check our answers, we can substitute the values of x back into the original equation:
For x = -1/3:
|3(-1/3) + 5| = 12(-1/3) + 8
|-1 + 5| = -4 + 8
|4| = 4
The left side of the equation equals 4, and the right side equals 4. Hence, x = -1/3 is a valid solution.
For x = -13/15:
|3(-13/15) + 5| = 12(-13/15) + 8
|(-39)/15 + 5| = -52/15 + 8
|(-39 + 75)/15| = (-52 + 120)/15
|36/15| = 68/15
The left side of the equation equals 12/5, and the right side equals 68/15. Since these values are not equal, x = -13/15 is not a valid solution.
Therefore, the only solution to the equation is x = -1/3.