https://r2.erweima.ai/mathbot/image/1726266135667s8sb7nn5.jpg

Let's analyze and solve the inequality given in the image.

We are asked to solve the inequality:

\[
|2x + 3| < -\frac{1}{2}x + 1
\]

**Step 1: Breaking the absolute value inequality**
An inequality with absolute values can be split into two cases:

- Case 1: \( 2x + 3 < -\frac{1}{2}x + 1 \)
- Case 2: \( -(2x + 3) < -\frac{1}{2}x + 1 \)

Let's solve both cases separately.

### **Case 1:**
\[
2x + 3 < -\frac{1}{2}x + 1
\]

To solve this inequality, first eliminate the fraction by multiplying through by 2:
\[
4x + 6 < -x + 2
\]
Now, add \(x\) to both sides:
\[
5x + 6 < 2
\]
Next, subtract 6 from both sides:
\[
5x < -4
\]
Finally, divide by 5:
\[
x < -\frac{4}{5}
\]

### **Case 2:**
\[
-(2x + 3) < -\frac{1}{2}x + 1
\]
Simplify:
\[
-2x - 3 < -\frac{1}{2}x + 1
\]
Again, multiply through by 2 to eliminate the fraction:
\[
-4x - 6 < -x + 2
\]
Now, add \(x\) to both sides:
\[
-3x - 6 < 2
\]
Add 6 to both sides:
\[
-3x < 8
\]
Finally, divide by -3 (and remember to flip the inequality sign):
\[
x > -\frac{8}{3}
\]

### **Step 2: Combine both cases**

From **Case 1**, we have \(x < -\frac{4}{5}\), and from **Case 2**, we have \(x > -\frac{8}{3}\).

Thus, the solution is the intersection of these two inequalities, which is:

\[
-\frac{8}{3} < x < -\frac{4}{5}
\]

### **Step 3: Verifying the options**

- (a) The problem can indeed be split into two inequalities, so **this is true**.
- (b) The solution set is not \(\left( -\infty, -\frac{4}{5} \right)\), so this is **false**.
- (c) The solution set is \(\left( -\frac{8}{3}, -\frac{4}{5} \right)\), which is the correct answer, so **this is true**.
- (d) Since option (c) is correct, **this is false**.

### Final Answer:
The correct options are (a) and (c).

---

Do you want further details or explanations?

Here are five related questions for practice:
1. How do you solve inequalities involving absolute values in general?
2. What are the rules for handling inequalities when multiplying or dividing by a negative number?
3. How do you represent solution sets on a number line?
4. What happens when the absolute value expression equals zero in such inequalities?
5. How would the inequality change if the inequality sign were "greater than" instead of "less than"?

**Tip:** Always remember to flip the inequality sign when multiplying or dividing both sides by a negative number!

|2x + 3| < -(1/2)x + 1. Which of the following options are true?

Solve the absolute value inequality |2x + 3| < -(1/2)x + 1 step-by-step by breaking it into two inequalities and finding the solution set. Ideal for high school students studying algebra and inequalities.

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