The problem is to solve the inequality:

$|x + 1| \leq |3x + 5|$

To solve this inequality, we'll break it down into different cases based on the properties of absolute values.

Step 1: Consider the points where the expressions inside the absolute values change sign.

• $x + 1 = 0$ gives $x = -1$
• $3x + 5 = 0$ gives $x = -\frac{5}{3}$

So, we need to consider the following intervals based on these points:

1. $x < -\frac{5}{3}$
2. $-\frac{5}{3} \leq x < -1$
3. $x \geq -1$

Step 2: Solve the inequality in each interval.

Interval 1: $x < -\frac{5}{3}$

• $x + 1 < 0$ and $3x + 5 < 0$
• Therefore, $|x + 1| = -(x + 1) = -x - 1$
• $|3x + 5| = -(3x + 5) = -3x - 5$

The inequality becomes: $-x - 1 \leq -3x - 5$ Simplifying: $3x - x \leq -5 + 1$ $2x \leq -4$ $x \leq -2$

Since $x < -\frac{5}{3}$, and $x \leq -2$ also satisfies this condition, this is a valid solution in this interval.

Interval 2: $-\frac{5}{3} \leq x < -1$

• $x + 1 < 0$ and $3x + 5 \geq 0$
• Therefore, $|x + 1| = -(x + 1) = -x - 1$
• $|3x + 5| = 3x + 5$

The inequality becomes: $-x - 1 \leq 3x + 5$ Simplifying: $-1 - 5 \leq 3x + x$ $-6 \leq 4x$ $x \geq -\frac{3}{2}$

In this interval $-\frac{5}{3} \leq x < -1$, and since $x \geq -\frac{3}{2}$, the solution is $-\frac{3}{2} \leq x < -1$.

Interval 3: $x \geq -1$

• $x + 1 \geq 0$ and $3x + 5 \geq 0$
• Therefore, $|x + 1| = x + 1$
• $|3x + 5| = 3x + 5$

The inequality becomes: $x + 1 \leq 3x + 5$ Simplifying: $1 - 5 \leq 3x - x$ $-4 \leq 2x$ $x \geq -2$

Since $x \geq -1$ is our interval, $x \geq -1$ is the solution in this case.

Step 3: Combine the solutions from all intervals.

• From Interval 1: $x \leq -2$
• From Interval 2: $-\frac{3}{2} \leq x < -1$
• From Interval 3: $x \geq -1$

So the final solution is: $x \leq -2 \quad \text{or} \quad x \geq -1$

Would you like further details on any step, or do you have any questions?

Here are some related questions you might find helpful:

1. How do you solve inequalities involving absolute values?
2. What are the key differences between solving equations and inequalities with absolute values?
3. How does the sign of a variable affect the solution of an inequality?
4. Can absolute value inequalities have no solution? If so, under what conditions?
5. How can graphing the functions help in solving absolute value inequalities?

Tip: When solving absolute value inequalities, always consider breaking down the inequality into multiple cases based on the critical points where the expressions inside the absolute values are zero.