How do you graph and solve #|6x|>24#?

The absolute value function takes any 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.

#-24 > 6x > 24#

Divide each segment of the system of inequalities by #color(red)(6)# to solve for #x# while keeping the system balanced:

#-24/color(red)(6) > (6x)/color(red)(6) > 24/color(red)(6)#

#-4 > (color(red)(cancel(color(black)(6)))x)/cancel(color(red)(6)) > 4#

#-4 > x > 4#

Or

#x < -4# and #x > 4#

Or, in interval notation:

#(-oo, -4)# and #(4, +oo)#

To graph this we will draw vertical lines at #-4# and #4# on the horizontal axis.

The lines will be a dashed lines because the inequality operators do not contain an "or equal to" clause.

We will shade to the left and right of each line because of the the "less than" and "greater than" inequality operators:

To graph and solve the inequality \(|6x| > 24\), we need to consider two cases: when \(6x\) is positive and when \(6x\) is negative. Case 1: \(6x > 24\) Divide both sides by 6 to isolate \(x\): \(x > 4\) Case 2: \(-6x > 24\) Divide both sides by -6. Since we're dividing by a negative number, flip the inequality sign: \(x < -4\) Combining the results from both cases, we have: \(x < -4\) or \(x > 4\) To graph the solution, we represent the two inequalities on the number line. We mark an open circle at -4 and shade to the left to represent \(x < -4\), and we mark an open circle at 4 and shade to the right to represent \(x > 4\). The union of these shaded regions represents the solution to the inequality. So, the solution is \(x < -4\) or \(x > 4\), and the graph would show an open circle at -4 and at 4, with shading to the left of -4 and to the right of 4 on the number line.