Solve the following absolute value equation.
2|4𝑥 − 3| + 1 = 9
a) 𝑥 =−
7
4
,
1
4
b) 𝑥 =
7
4
c) 𝑥 = 2, −
1
2
d) 𝑥 =
7
4
, −
1
4
To solve the equation, we need to isolate the absolute value term and then solve for 𝑥.
Starting with the given equation: 2|4𝑥 − 3| + 1 = 9
Subtracting 1 from both sides: 2|4𝑥 − 3| = 8
Dividing both sides by 2: |4𝑥 − 3| = 4
We now have two cases to consider:
Case 1: (4𝑥 − 3) is positive
In this case, the absolute value bars can be removed and we have: 4𝑥 − 3 = 4
Adding 3 to both sides: 4𝑥 = 7
Dividing both sides by 4: 𝑥 = 7/4
Case 2: (4𝑥 − 3) is negative
In this case, we need to change the sign inside the absolute value: |-(4𝑥 − 3)| = 4
Simplifying the absolute value expression: |-4𝑥 + 3| = 4
Since the expression inside the absolute value bars is now negative, we need to remove the negative sign: 4𝑥 − 3 = 4
Adding 3 to both sides: 4𝑥 = 7
Dividing both sides by 4: 𝑥 = 7/4
Thus, the solution to the absolute value equation is 𝑥 = 7/4.
Therefore, the correct option is: b) 𝑥 = 7/4.
To solve the absolute value equation 2|4𝑥 − 3| + 1 = 9, we need to isolate the absolute value term and then remove the absolute value by considering both the positive and negative cases.
1. Subtract 1 from both sides of the equation:
2|4𝑥 − 3| = 8
2. Divide both sides of the equation by 2:
|4𝑥 − 3| = 4
3. Now we consider both the positive and negative cases of the absolute value.
Positive Case:
4𝑥 − 3 = 4
Solve for 𝑥:
4𝑥 = 7
𝑥 = 7/4
Negative Case:
4𝑥 − 3 = -4
Solve for 𝑥:
4𝑥 = -1
𝑥 = -1/4
So, the solutions to the absolute value equation 2|4𝑥 − 3| + 1 = 9 are 𝑥 = 7/4 and 𝑥 = -1/4.
Therefore, the answer is d) 𝑥 = 7/4, -1/4.
To solve the absolute value equation 2|4𝑥 − 3| + 1 = 9, we need to isolate the absolute value expression and then solve for 𝑥. Here's how you can solve it step by step:
Step 1: Subtract 1 from both sides of the equation to isolate the absolute value expression:
2|4𝑥 − 3| = 8
Step 2: Divide both sides of the equation by 2 to isolate the absolute value expression:
|4𝑥 − 3| = 4
Step 3: Break the equation into two separate equations, removing the absolute value notation:
4𝑥 − 3 = 4
and
4𝑥 − 3 = -4
Step 4: Solve the first equation:
4𝑥 − 3 = 4
Add 3 to both sides:
4𝑥 = 7
Divide both sides by 4:
𝑥 = 7/4
Step 5: Solve the second equation:
4𝑥 − 3 = -4
Add 3 to both sides:
4𝑥 = -1
Divide both sides by 4:
𝑥 = -1/4
So the possible solutions for the given absolute value equation are:
a) 𝑥 = 7/4 (Option d)
b) 𝑥 = -1/4 (Option d)
Therefore, the correct answer is d) 𝑥 = 7/4, -1/4.