How many integers are in the solution set `|2x+6|lt (19)/(2)` ?

Options

A. None
B. Two
C. Nine
D. Fourteen

Explanation

To solve the inequality \( |2x + 6| < \frac{19}{2} \), we will follow these steps:

### Step 1: Understand the Absolute Value Inequality
The expression \( |A| < B \) implies that \( -B < A < B \). Therefore, we can rewrite the given inequality as:
\[ -\frac{19}{2} < 2x + 6 < \frac{19}{2} \]

### Step 2: Split into Two Inequalities
This gives us two separate inequalities to solve:
1. \( 2x + 6 < \frac{19}{2} \)
2. \( 2x + 6 > -\frac{19}{2} \)

### Step 3: Solve the First Inequality
Starting with the first inequality:
\[ 2x + 6 < \frac{19}{2} \]
Subtract 6 from both sides:
\[ 2x < \frac{19}{2} - 6 \]
Convert 6 to a fraction with a denominator of 2:
\[ 6 = \frac{12}{2} \]
Now, substitute:
\[ 2x < \frac{19}{2} - \frac{12}{2} = \frac{7}{2} \]
Now, divide both sides by 2:
\[ x < \frac{7}{4} \]

### Step 4: Solve the Second Inequality
Now, for the second inequality:
\[ 2x + 6 > -\frac{19}{2} \]
Subtract 6 from both sides:
\[ 2x > -\frac{19}{2} - 6 \]
Convert 6 to a fraction:
\[ 6 = \frac{12}{2} \]
Now, substitute:
\[ 2x > -\frac{19}{2} - \frac{12}{2} = -\frac{31}{2} \]
Now, divide both sides by 2:
\[ x > -\frac{31}{4} \]

### Step 5: Combine the Results
Now we have:
\[ -\frac{31}{4} < x < \frac{7}{4} \]
Converting these to decimal form:
\[ -\frac{31}{4} = -7.75 \quad \text{and} \quad \frac{7}{4} = 1.75 \]
Thus, the inequality can be expressed as:
\[ -7.75 < x < 1.75 \]

### Step 6: Identify the Integer Solutions
Now, we need to find the integers that lie between -7.75 and 1.75. The integers in this range are:
-7, -6, -5, -4, -3, -2, -1, 0, 1

### Step 7: Count the Integers
Counting these integers gives us:
-7, -6, -5, -4, -3, -2, -1, 0, 1 → Total: 9 integers.

### Final Answer
The number of integers in the solution set is **9**.

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