Question

solve for x and graph the solution.
28 < |x - 23|
click two endpoints to graph a line segment, an endpoint and an arrowhead to graph a ray, or two arrowheads to graph a line. to change endpoints from filled - in circles to empty circles, click on them.
(then there is a number line with marks at - 40, - 30, - 20, - 10, 0, 10, 20, 30, 40, 50, 60, 70, 80 and an arrow at the end pointing right.)

Explanation:

Step1: Recall absolute value inequality rule

The inequality \( 23 < |s - 23| \) means \( |s - 23|>23 \). By the definition of absolute value, if \( |x|>a \) (\( a>0 \)), then \( x < -a \) or \( x > a \). Here, \( x = s - 23 \) and \( a = 23 \), so we have two cases:

Case 1: \( s - 23 < - 23 \)

Step2: Solve the first inequality

Add 23 to both sides of \( s - 23 < - 23 \):
\( s - 23+23 < - 23 + 23 \)
Simplify to get \( s < 0 \).

Case 2: \( s - 23 > 23 \)

Step3: Solve the second inequality

Add 23 to both sides of \( s - 23 > 23 \):
\( s - 23+23 > 23 + 23 \)
Simplify to get \( s > 46 \).

For graphing:

• For \( s < 0 \), we draw an open circle at \( 0 \) (since \( s = 0 \) is not a solution) and an arrow pointing to the left (towards negative infinity).
• For \( s > 46 \), we draw an open circle at \( 46 \) (since \( s = 46 \) is not a solution) and an arrow pointing to the right (towards positive infinity).

Answer:

\( s < 0 \) or \( s > 46 \)