Question

solve for w and graph the solution.\\(|w - 4| > 2\\)\\(\\)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.\\(\\)\\(\\) (and the number line with marks at -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10)

Explanation:

Step1: Solve the absolute - value inequality

The absolute - value inequality is \(|w - 4|>2\). By the definition of absolute value, if \(|x|>a\) (\(a>0\)), then \(x < - a\) or \(x>a\). Here, \(x = w - 4\) and \(a = 2\). So we have two cases:
Case 1: \(w - 4>2\)
Add 4 to both sides of the inequality: \(w-4 + 4>2 + 4\), which simplifies to \(w>6\).
Case 2: \(w - 4<-2\)
Add 4 to both sides of the inequality: \(w-4 + 4<-2 + 4\), which simplifies to \(w<2\).

Step2: Graph the solution

• For \(w < 2\): On the number line, we draw an open circle at \(w = 2\) (because the inequality is strict, \(w

eq2\)) and draw an arrow pointing to the left (towards negative infinity) to represent all values of \(w\) less than 2.

• For \(w>6\): On the number line, we draw an open circle at \(w = 6\) (because the inequality is strict, \(w

eq6\)) and draw an arrow pointing to the right (towards positive infinity) to represent all values of \(w\) greater than 6.

Answer:

\( w < 2 \) or \( w > 6 \)