How many integers are there in the solution set of `|2x+6|lt19/2`?

Options

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

Answer

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

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

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

### Step 3: Solve the First Inequality
For 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, substituting:
\[ 2x > -\frac{19}{2} - \frac{12}{2} = -\frac{31}{2} \]
Now, divide both sides by 2:
\[ x > -\frac{31}{4} \]

### Step 4: Solve the Second Inequality
For the second 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, substituting:
\[ 2x < \frac{19}{2} - \frac{12}{2} = \frac{7}{2} \]
Now, divide both sides by 2:
\[ x < \frac{7}{4} \]

### Step 5: Combine the Results
Now we have the combined inequality:
\[ -\frac{31}{4} < x < \frac{7}{4} \]

### Step 6: Identify the Integer Solutions
Now we need to find the integers that lie within the interval \( -\frac{31}{4} \) and \( \frac{7}{4} \).

Calculating the values:
- \( -\frac{31}{4} = -7.75 \) (so the smallest integer greater than this is -7)
- \( \frac{7}{4} = 1.75 \) (so the largest integer less than this is 1)

The integers in the interval \( -7 < x < 1.75 \) 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

Thus, there are a total of **9 integers** in the solution set.

### Final Answer
The number of integers in the solution set of \( |2x + 6| < \frac{19}{2} \) is **9**.

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