How do you solve and graph #abs(5-x)=3#?

Question

How do you solve and graph #abs(5-x)>=3#?

Eva Abramson · Accepted Answer

See a solution process below:

The absolute value function takes any negative or positive term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent. #-3 >= 5 - x >= 3# First, subtract #color(red)(5)# from each segment of the system of inequalities to isolate the #x# term while keeping the system balanced: #-color(red)(5) - 3 >= -color(red)(5) + 5 - x >= -color(red)(5) + 3# #-8 >= 0 - x >= -2# #-8 >= -x >= -2# Now, multiply each segment by #color(blue)(-1)# to solve for #x# while keeping the system balanced. However, because we are multiplying or dividing inequalities by a negative number we must reverse the inequality operators: #color(blue)(-1) xx -8 color(red)(<=) color(blue)(-1) xx -x color(red)(<=) color(blue)(-1) xx -2# #8 color(red)(<=) x color(red)(<=) 2# Or #x <= 2#; #x >= 8# Or, in interval notation: #(-oo, 2]#; #[8, +oo)#

Jace Anderson · Answer

undefined To solve and graph $|5-x| \geq 3$, follow these steps:

1. Split the absolute value inequality into two separate inequalities:
   a) $5 - x \geq 3$
   b) $5 - x \leq -3$

2. Solve each inequality separately for $x$.

3. Graph the solutions on a number line.

4. Shade the regions that satisfy the original inequality.

The solution and graph for $|5-x| \geq 3$ are as follows:

- For $5 - x \geq 3$, solving for $x$ gives $x \leq 2$.
- For $5 - x \leq -3$, solving for $x$ gives $x \geq 8$.

Graphically, this represents shading the regions to the left of $x = 2$ and to the right of $x = 8$ on the number line, including the endpoints.

How do you solve and graph #abs(5-x)&gt;=3#?

How do you solve and graph #abs(5-x)>=3#?