Suppose 3 < *a* < 4 and 4 < *b* < 5. Find all possible values of each expression. -2a+b

Given the inequalities:
\[
3 < a < 4 \quad \text{and} \quad 4 < b < 5,
\]
we are tasked with finding the possible values of the expression:
\[
-2a + b.
\]

### Step 1: Analyze the bounds for \( a \) and \( b \)
- For \( a \), we know that \( 3 < a < 4 \).
- For \( b \), we know that \( 4 < b < 5 \).

### Step 2: Express the bounds of \( -2a \)
Multiplying the inequality for \( a \) by \(-2\), and remembering to reverse the inequality when multiplying by a negative number:
\[
-2(3) > -2a > -2(4),
\]
\[
-6 > -2a > -8.
\]
Thus, the range for \( -2a \) is:
\[
-8 < -2a < -6.
\]

### Step 3: Combine with \( b \)
We now consider the full expression:
\[
-2a + b.
\]
Since \( -8 < -2a < -6 \) and \( 4 < b < 5 \), adding \( b \) to \( -2a \) gives:
\[
-8 + 4 < -2a + b < -6 + 5,
\]
\[
-4 < -2a + b < -1.
\]

### Conclusion:
The possible values of the expression \( -2a + b \) lie in the range:
\[
-4 < -2a + b < -1.
\]

Would you like a more detailed explanation or have any questions?

Here are 5 related questions for further practice:
1. What is the range of \( a^2 + b \) for the given values of \( a \) and \( b \)?
2. If \( 2 < c < 3 \), what is the range of \( a + b - c \)?
3. What is the range of \( ab \) given \( 3 < a < 4 \) and \( 4 < b < 5 \)?
4. How would the range of \( -2a + b \) change if \( a \) were instead \( 2 < a < 3 \)?
5. What is the range of \( b/a \) for the given values of \( a \) and \( b \)?

**Tip:** When multiplying or dividing inequalities by negative numbers, always remember to reverse the inequality sign.

Suppose 3 < a < 4 and 4 < b < 5. Find all possible values of each expression. -2a + b

Learn how to find the range of the expression -2a + b given the inequalities 3 < a < 4 and 4 < b < 5. This problem covers key algebraic concepts such as working with inequalities and manipulating algebraic expressions. Suitable for students in Grades 9-10.

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