Let's solve the three inequalities provided in the image one by one. They are all absolute value inequalities.

1. $5|x + 3| - 1 \leq 9$

Step 1: Start by isolating the absolute value expression: $5|x + 3| - 1 \leq 9$ Add 1 to both sides: $5|x + 3| \leq 10$ Now divide both sides by 5: $|x + 3| \leq 2$

Step 2: Break the absolute value inequality into two cases: $-2 \leq x + 3 \leq 2$

Step 3: Solve for $x$: $-2 - 3 \leq x \leq 2 - 3$ $-5 \leq x \leq -1$

So, the solution to the first inequality is: $x \in [-5, -1]$

2. $4|2x + 3| > 12$

Step 1: Start by isolating the absolute value expression: $4|2x + 3| > 12$ Divide both sides by 4: $|2x + 3| > 3$

Step 2: Break the absolute value inequality into two cases: $2x + 3 > 3 \quad \text{or} \quad 2x + 3 < -3$

Step 3: Solve for $x$ in both cases:

• Case 1: $2x + 3 > 3$ $2x > 0 \quad \Rightarrow \quad x > 0$
• Case 2: $2x + 3 < -3$ $2x < -6 \quad \Rightarrow \quad x < -3$

So, the solution to the second inequality is: $x > 0 \quad \text{or} \quad x < -3$ This is written as $x \in (-\infty, -3) \cup (0, \infty)$.

3. $-3|x + 5| > 15$

Step 1: Start by isolating the absolute value expression: $-3|x + 5| > 15$ Divide both sides by -3, and remember to flip the inequality: $|x + 5| < -5$

Step 2: Notice that absolute values are always non-negative, so there are no real solutions where the absolute value is less than a negative number.

So, the third inequality has no solution.

Final Answer:

• For the first inequality: $x \in [-5, -1]$
• For the second inequality: $x \in (-\infty, -3) \cup (0, \infty)$
• For the third inequality: no solution

Number Line:

• For the first inequality, there will be a closed circle at $x = -5$ and $x = -1$, with a solid line between them.
• For the second inequality, there will be open circles at $x = -3$ and $x = 0$, with solid lines extending to negative and positive infinity.

Would you like me to sketch the number line, or provide further clarification on any of the steps?

Here are 5 follow-up questions:

1. How do you solve absolute value inequalities involving both "greater than" and "less than" symbols?
2. What are the different ways to represent solutions to inequalities?
3. Why do you flip the inequality sign when dividing by a negative number?
4. How can you graph the solution set for an absolute value inequality on a number line?
5. What are common situations where inequalities result in "no solution"?

Tip: Always check whether the inequality conditions are possible before solving, as sometimes inequalities can have no solution based on the absolute value constraints.