How do you solve #| 1/2-3/4x | + 1=3 #?

Question

Savannah Anderson · Accepted Answer

#-#2 and 10/3.

The given equation is equivalent to the pair 1/2#-#3 x/4 = #+-#2.

Abigail Allen · Answer

#x = -2" or "x=10/3" "#Note that #10/3 = 3.33bar3#

The very nature of this type of problem is that it usually has 2 solutions for #x#

We need to end up with the 'Absolute' part of the equation on its own and on the left of the equals sign and everything else on the other.

Given:#" "color(brown)(|1/2-3/4 x|+1=3)#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Simplifying")#

Subtract #color(blue)(1)# from both sides

#" "color(brown)(|1/2-3/4 x|+1color(blue)(-1)" "=" "3color(blue)(-1))#

#" "|1/2-3/4 x|+0" "=" "2#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Solving for "x)#

Think of this as:

#" "|1/2-3/4 x|" "=" "|+-2|#

#color(brown)("Consider: "1/2-3/4 x" "=" "-2" "#

Multiply by (-1 ) giving

#" "-1/2+3/4 x=+2#

#" "3/4 x=2+1/2#

#color(blue)(" " x" "=" "4/3xx5/2" " = " "10/3)#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(brown)("Consider: "1/2-3/4 x=+2#

Multiply by (-1 ) giving

#" "-1/2+3/4 x=-2#

#" "3/4 x=-2+1/2#

#" "color(blue)(x=4/3xx(-3/2) = -2)#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#" "color(magenta)(x = -2" or "x=10/3)#

Eli Assouline · Answer

undefined To solve the equation $| \frac{1}{2} - \frac{3}{4}x | + 1 = 3$:

1. Subtract 1 from both sides to isolate the absolute value term: $| \frac{1}{2} - \frac{3}{4}x | = 2$.
2. Since the absolute value of a number can be positive or negative, we set up two equations:
   a. $\frac{1}{2} - \frac{3}{4}x = 2$
   b. $\frac{1}{2} - \frac{3}{4}x = -2$.
3. Solve each equation separately for $x$.

Solving equation (a):
1. Subtract $\frac{1}{2}$ from both sides: $-\frac{3}{4}x = \frac{3}{2}$.
2. Multiply both sides by $-\frac{4}{3}$ to solve for $x$: $x = -2$.

Solving equation (b):
1. Subtract $\frac{1}{2}$ from both sides: $-\frac{3}{4}x = -\frac{5}{2}$.
2. Multiply both sides by $-\frac{4}{3}$ to solve for $x$: $x = \frac{10}{3}$.

So, the solutions to the equation are $x = -2$ and $x = \frac{10}{3}$.