To solve the inequality \(-2 < 7 - 3x \leq 10\), we can break it into two parts and solve each part separately.
- Solving the first part:
\[ -2 < 7 - 3x \]
Subtract \(7\) from both sides:
\[ -2 - 7 < -3x \quad \Rightarrow \quad -9 < -3x \]
Now, divide both sides by \(-3\). Remember that dividing by a negative number reverses the inequality:
\[ \frac{-9}{-3} > x \quad \Rightarrow \quad 3 > x \quad \Rightarrow \quad x < 3 \]
- Solving the second part:
\[ 7 - 3x \leq 10 \]
Subtract \(7\) from both sides:
\[ -3x \leq 10 - 7 \quad \Rightarrow \quad -3x \leq 3 \]
Now, divide both sides by \(-3\) (again, reversing the inequality):
\[ x \geq \frac{3}{-3} \quad \Rightarrow \quad x \geq -1 \]
- Combining the results from both parts, we have:
\[ -1 \leq x < 3 \]
This means the integral values of \(x\) that satisfy the compound inequality are \(-1, 0, 1, 2\).
Thus, the integral values that satisfy the inequality \(-2 < 7 - 3x \leq 10\) are:
\(-1, 0, 1, 2\)