We are given the inequality:

$-5x + 8 < 14$

and we are tasked with finding the integer values of $x$ in the interval $[-3, 3]$ that satisfy this inequality.

Step 1: Solve the inequality

Start by isolating $x$:

$-5x + 8 < 14$

Subtract 8 from both sides:

$-5x < 14 - 8$

$-5x < 6$

Now, divide both sides by $-5$, remembering to flip the inequality sign (since dividing by a negative reverses the inequality):

$x > \frac{6}{-5}$

$x > -1.2$

Step 2: Find integer values in the interval $[-3, 3]$

We are looking for integer values of $x$ that are both greater than $-1.2$ and within the interval $[-3, 3]$. The integers greater than $-1.2$ and less than or equal to 3 are:

$x = -1, 0, 1, 2, 3$

Final Answer:

The integer values of $x$ that satisfy the inequality are:

$x = -1, 0, 1, 2, 3$

Do you need more details on this? Here are five related questions you can explore:

1. How do you solve inequalities that involve multiplying or dividing by negative numbers?
2. What happens to the inequality sign when you divide by a negative number?
3. How do you graph solutions of inequalities on a number line?
4. Can you explain the importance of integer constraints in inequality problems?
5. How do you handle compound inequalities with multiple conditions?

Tip: Always remember to flip the inequality sign when dividing or multiplying both sides of an inequality by a negative number.