How do you solve absolute value equation #4abs(2y - 7) + 5 = 9#?

Question

Hazel Anglin · Accepted Answer

There are several ways to approach this.  One method which tends to avoid extraneous results is to isolate the absolute value on one side of the equation, square both sides, and solve the resulting quadratic equation. #4abs(2y-7) + 5 = 9#
implies
#abs(2y-7) = 1# #(2y-7)^2 = 1# #4y^2 -28y +49 = 1# #y^2 -7y +12 = 0# by factoring
#(y-3)(y-4)=0# therefore
#y=3#
or
#y=4#

Ethan Adkins · Answer

Absolute value equations usually have two solutions First we simplify, we subtract 5 on both sides and then divide by 4: #4|2y-7|+5=9->|2y-7|=1# For #y>3 1/2->2y-7=1->y=4# For #y<3 1/2 ->7-2y=1->y=3# For #y=3 1/2# there is no solution.
 graph{|2x-7| [-1.19, 11.304, -1.313, 4.93]}

Savannah Anderson · Answer

undefined To solve the absolute value equation $4|2y - 7| + 5 = 9$:
1. Subtract 5 from both sides to isolate the absolute value term: $4|2y - 7| = 4$.
2. Divide both sides by 4: $|2y - 7| = 1$.
3. Set up two equations based on the positive and negative cases of the absolute value:
   a. For the positive case: $2y - 7 = 1$.
   b. For the negative case: $2y - 7 = -1$.
4. Solve each equation separately:
   a. $2y - 7 = 1$ leads to $2y = 8$ and $y = 4$.
   b. $2y - 7 = -1$ leads to $2y = 6$ and $y = 3$.
5. The solutions are $y = 4$ and $y = 3$.

Hazel Anglin · Answer

undefined To solve the absolute value equation $4|2y - 7| + 5 = 9$, follow these steps:

1. Subtract 5 from both sides of the equation:
   $4|2y - 7| = 9 - 5$
   $4|2y - 7| = 4$

2. Divide both sides of the equation by 4:
   $|2y - 7| = \frac{4}{4}$
   $|2y - 7| = 1$

3. Now, split the equation into two cases:
   $2y - 7 = 1$ or $2y - 7 = -1$

4. Solve each case separately:
   Case 1:
   $2y - 7 = 1$
   $2y = 1 + 7$
   $2y = 8$
   $y = \frac{8}{2}$
   $y = 4$

Case 2:
   $2y - 7 = -1$
   $2y = -1 + 7$
   $2y = 6$
   $y = \frac{6}{2}$
   $y = 3$

So, the solutions to the equation are $y = 4$ and $y = 3$.