Question

solve for s and graph the solution.
106 ≤ |4s - 540| + 6
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.

Explanation:

Step1: Isolate the absolute - value expression

Subtract 6 from both sides of the inequality \(106\leq|4s - 540|+6\).
\(106−6\leq|4s - 540|+6−6\)
\(100\leq|4s - 540|\)
Since \(|x|\geq a\) is equivalent to \(x\geq a\) or \(x\leq - a\) (where \(a = 100\) and \(x = 4s-540\)), we have two inequalities:
\(4s - 540\geq100\) or \(4s - 540\leq - 100\)

Step2: Solve the first inequality \(4s - 540\geq100\)

Add 540 to both sides: \(4s-540 + 540\geq100 + 540\)
\(4s\geq640\)
Divide both sides by 4: \(s\geq\frac{640}{4}\)
\(s\geq160\)

Step3: Solve the second inequality \(4s - 540\leq - 100\)

Add 540 to both sides: \(4s-540 + 540\leq - 100+540\)
\(4s\leq440\)
Divide both sides by 4: \(s\leq\frac{440}{4}\)
\(s\leq110\)

Answer:

The solution of the inequality is \(s\leq110\) or \(s\geq160\). On the number - line, we graph a ray starting at \(s = 110\) (with a filled - in circle) and going to the left, and a ray starting at \(s = 160\) (with a filled - in circle) and going to the right.