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.