Solve the inequality and graph the solution.

To solve the inequality \(4z - 6 > 2\), follow these steps:

1. Add 6 to both sides to isolate the term with \(z\): \[ 4z - 6 + 6 > 2 + 6 \] This simplifies to: \[ 4z > 8 \]

2. Divide both sides by 4 to solve for \(z\): \[ z > \frac{8}{4} \] This simplifies to: \[ z > 2 \]

Solution

The solution to the inequality is: \[ z > 2 \]

Graphing the Solution

To graph this solution on a number line:

• Draw a number line.
• Mark the point \(2\) with an open circle to indicate that \(2\) is not included in the solution (since \(z\) must be greater than \(2\)).
• Shade the region to the right of the point \(2\) to indicate that all numbers greater than \(2\) are included in the solution.

Here's a simple representation of the number line:

<---|----|----|----|----|----|----|----|--->
    0    1    2    3    4    5    6
         (----------------->

The open circle at \(2\) signifies that \(2\) is not included in the solution, and the arrow indicates that the solution continues indefinitely to the right.