Solve the following compound inequalities. Use both a line graph and interval notation to write each solution set.4⁢𝑥-3<-2   or   4⁢𝑥-3>2

Sure, let's solve these inequalities step by step.

First, let's solve the inequality 4x - 3 < -2:

Step 1: Add 3 to both sides of the inequality to isolate the term with x on one side: 4x - 3 + 3 < -2 + 3 This simplifies to: 4x < 1

Step 2: Divide both sides of the inequality by 4 to solve for x: 4x / 4 < 1 / 4 This simplifies to: x < 1/4

Now, let's solve the inequality 4x - 3 > 2:

Step 1: Add 3 to both sides of the inequality to isolate the term with x on one side: 4x - 3 + 3 > 2 + 3 This simplifies to: 4x > 5

Step 2: Divide both sides of the inequality by 4 to solve for x: 4x / 4 > 5 / 4 This simplifies to: x > 5/4

So, the solution to the compound inequality is x < 1/4 or x > 5/4.

In interval notation, this is written as (-∞, 1/4) U (5/4, ∞).

On a line graph, you would draw two rays: one starting at 1/4 and extending to the left (indicating x < 1/4), and one starting at 5/4 and extending to the right (indicating x > 5/4). Both points, 1/4 and 5/4, would be open circles on the graph, indicating that these values are not included in the solution set.