How do you solve #-3+ 4| 8n + 10| = 53#?

Question

Isaiah Anthony · Accepted Answer

See the entire solution process below.

First, we need to solve for the absolute value term. To start, add #color(red)(3)# to each side of the equation to eliminate the #-3# term on the left side of the equation while keeping the equation balanced: #color(red)(3) - 3 + 4abs(8n + 10) = color(red)(3) + 53# #0 + 4abs(8n + 10) = 56# #4abs(8n + 10) = 56# Next, divide each side of the equation by #color(red)(4)# to solve for the absolute value function while keeping the equation balanced: #(4abs(8n + 10))/color(red)(4) = 56/color(red)(4)# #(color(red)(cancel(color(black)(4)))abs(8n + 10))/cancel(color(red)(4)) = 14# #abs(8n + 10) = 14# The absolute value function takes any negative or positive term and transforms it to its positive form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent. Solution 1) #8n + 10 = -14# #8n + 10 - color(red)(10) = -14 - color(red)(10)# #8n + 0 = -24# #8n = -24# #(8n)/color(red)(8) = -24/color(red)(8)# #(color(red)(cancel(color(black)(8)))n)/cancel(color(red)(8)) = -3# #n = -3# Solution 2) #8n + 10 = 14# #8n + 10 - color(red)(10) = 14 - color(red)(10)# #8n + 0 = 4# #8n = 4# #(8n)/color(red)(8) = 4/color(red)(8)# #(color(red)(cancel(color(black)(8)))n)/cancel(color(red)(8)) = 1/2# #n = 1/2# The solution is: #n = -3# and #n = 1/2#

Isaiah Anthony · Answer

undefined To solve the equation $-3 + 4|8n + 10| = 53$, follow these steps:

1. First, isolate the absolute value term by adding 3 to both sides:
   $$4|8n + 10| = 56$$

2. Next, divide both sides by 4 to isolate the absolute value:
   $$|8n + 10| = 14$$

3. Now, break the equation into two cases:
   Case 1: $8n + 10 = 14$
   Case 2: $8n + 10 = -14$

4. For Case 1, solve for $n$:
   $$8n + 10 = 14$$
   $$8n = 4$$
   $$n = \frac{4}{8}$$
   $$n = \frac{1}{2}$$

5. For Case 2, solve for $n$:
   $$8n + 10 = -14$$
   $$8n = -24$$
   $$n = \frac{-24}{8}$$
   $$n = -3$$

6. So, the solutions for the equation are $n = \frac{1}{2}$ and $n = -3$.