Let's analyze the first inequality as an example to find the five points and the solution set. The process will be similar for the others.

1) $y > 2 |x - 3|$

Step 1: Understanding the inequality

We need to graph the inequality $y > 2 |x - 3|$, which is a transformation of the basic absolute value function $y = |x|$. The expression $|x - 3|$ shifts the graph 3 units to the right, and multiplying by 2 stretches the graph vertically by a factor of 2.

Step 2: Finding 5 points

We pick values of $x$, substitute them into the equation $y = 2|x - 3|$, and then plot these points.

• When $x = 1$: $y = 2|1 - 3| = 2(2) = 4$ Point: $(1, 4)$

• When $x = 2$: $y = 2|2 - 3| = 2(1) = 2$ Point: $(2, 2)$

• When $x = 3$: $y = 2|3 - 3| = 2(0) = 0$ Point: $(3, 0)$

• When $x = 4$: $y = 2|4 - 3| = 2(1) = 2$ Point: $(4, 2)$

• When $x = 5$: $y = 2|5 - 3| = 2(2) = 4$ Point: $(5, 4)$

Step 3: Graphing the inequality

Once these points are plotted, we graph the line and then shade the area above the line because the inequality is $y > 2 |x - 3|$. This shaded region represents the solution set.

Would you like to go through another inequality from the list or need more details on this one?

Follow-up Questions:

1. How does the transformation affect the graph of an absolute value function?
2. What happens if we have a "less than" inequality instead of "greater than"?
3. How do vertical stretches or compressions affect the steepness of the graph?
4. What is the difference between $y = |x|$ and $y = |x - h|$?
5. How does shifting horizontally or vertically change the solution set?

Tip:

When graphing absolute value inequalities, always plot the "equal to" line first and then determine the region to shade based on the inequality symbol.