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Graphing inequalities involves plotting regions on a coordinate plane that satisfy the given inequality. Inequalities describe a range of values rather than exact points. To graph an inequality, you follow these steps:
1. Graph the boundary line or curve. The inequality symbol determines if you need a solid line (≤ or ≥) or a dashed line (< or >).
2. Determine which side of the boundary the inequality covers by selecting a test point and substituting it into the inequality.
3. Shade the region containing all the points that satisfy the inequality.
For instance, in the inequality system given in the exercise:

• For the inequality **y ≤ log x**, you graph the curve of the logarithmic function **y = log x** and shade below the curve.
• For the inequality **y ≥ |x - 2|**, you plot the V-shaped absolute value function **y = |x - 2|** and shade above this V-shape.

The final shaded region represents the solution set where both inequalities overlap.

Logarithmic functions are essential in understanding exponential growth and decay. The function **y = log x** has some unique properties:
1. The logarithm base is usually 10 (common logarithm) or e (natural logarithm). When not specified, we often assume the base of 10.
2. The curve **y = log x** passes through the point **(1, 0)** because **log(1) = 0**.
3. For values **x > 1**, **y = log x** is positive, and for **0 < x < 1**, **y = log x** is negative.
When graphing logarithmic inequalities like **y ≤ log x**, remember:

• The function only exists for **x > 0**.
• **y** values become negative as **x** approaches 0 from the right and grow larger as **x** increases.
• When shading for **y ≤ log x**, you shade all the area below the curve of **y = log x**.

Understanding these ideas will help you master graphing logarithmic inequalities.

Absolute value functions depict distances and always produce non-negative results. The standard form is **y = |x - h|**, where **h** is a horizontal shift. To graph **y = |x - 2|**, follow these guidelines:
1. Identify the vertex of the absolute value function, which will be at **(2, 0)** for **y = |x - 2|**.
2. Recognize the V-shape centered at the vertex. For **x > 2**, the function grows linearly with a slope of 1 (**y = x - 2**), and for **x < 2**, it decreases linearly (**y = 2 - x**).
When dealing with the inequality **y ≥ |x - 2|**:

• Graph the boundary line **y = |x - 2|** with a solid line since the inequality includes equal to (≥).
• Shade the region above the V-shape because **y** values are greater than or equal to the absolute value function.

This region represents all the points satisfying **y ≥ |x - 2|**. Combining this with the previous logarithmic function graph will help you find the overlapping region which represents the complete solution to the system of inequalities.