If `2x^(2)+3ax-7lt-17`, what is the smallest possible integer value of a when `x=-1`?

Answer

To solve the inequality \( 2x^2 + 3ax - 7 < -17 \) for the smallest possible integer value of \( a \) when \( x = -1 \), we will follow these steps:

### Step 1: Substitute \( x = -1 \) into the inequality
We start by substituting \( x = -1 \) into the inequality:

\[ 2(-1)^2 + 3a(-1) - 7 < -17 \]

### Step 2: Simplify the expression
Calculate \( 2(-1)^2 \):

\[ 2(1) = 2 \]

Now substitute this back into the inequality:

\[ 2 + 3a(-1) - 7 < -17 \]

This simplifies to:

\[ 2 - 3a - 7 < -17 \]

### Step 3: Combine like terms
Combine the constants \( 2 - 7 \):

\[ -3a - 5 < -17 \]

### Step 4: Isolate the term with \( a \)
Add 5 to both sides of the inequality:

\[ -3a < -17 + 5 \]

This simplifies to:

\[ -3a < -12 \]

### Step 5: Divide by -3
When dividing by a negative number, we reverse the inequality sign:

\[ a > 4 \]

### Step 6: Find the smallest integer value of \( a \)
Since \( a \) must be greater than 4, the smallest possible integer value for \( a \) is:

\[ a = 5 \]

### Final Answer
The smallest possible integer value of \( a \) is \( 5 \).

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