Question

1. the inequality $5x + 8y \leq 2400$ represents possible combinations of chairs (x) and tables (y) that can fit in a banquet hall. which combinations are possible? select all that apply.\

\\(\square\\) a. 480 chairs and 0 tables\
\\(\square\\) b. 0 chairs and 300 tables\
\\(\square\\) c. 200 chairs and 150 tables\
\\(\square\\) d. 300 chairs and 100 tables\
\\(\square\\) e. 400 chairs and 50 tables\
\\(\square\\) f. 100 chairs and 250 tables

Explanation:

Step1: Check Option A

Substitute \(x = 480\), \(y = 0\) into \(5x + 8y\):
\(5(480)+8(0)=2400 + 0 = 2400\). Since \(2400\leq2400\), this is valid.

Step2: Check Option B

Substitute \(x = 0\), \(y = 300\) into \(5x + 8y\):
\(5(0)+8(300)=0 + 2400 = 2400\). Since \(2400\leq2400\), this is valid? Wait, no: \(8\times300 = 2400\), but the inequality is \(5x + 8y\leq2400\). Wait, \(5(0)+8(300)=2400\), which is equal, so it should be valid? Wait, no, let's recalculate: \(8\times300 = 2400\), so \(5(0)+8(300)=2400\), which satisfies \(2400\leq2400\). Wait, but let's check again. Wait, the inequality is \(5x + 8y\leq2400\). For option B: \(x = 0\), \(y = 300\). So \(5(0)+8(300)=2400\), which is equal to 2400, so it's valid. Wait, but let's check other options.

Step3: Check Option C

Substitute \(x = 200\), \(y = 150\) into \(5x + 8y\):
\(5(200)+8(150)=1000 + 1200 = 2200\). Since \(2200\leq2400\), this is valid.

Step4: Check Option D

Substitute \(x = 300\), \(y = 100\) into \(5x + 8y\):
\(5(300)+8(100)=1500 + 800 = 2300\). Since \(2300\leq2400\), this is valid.

Step5: Check Option E

Substitute \(x = 400\), \(y = 50\) into \(5x + 8y\):
\(5(400)+8(50)=2000 + 400 = 2400\). Since \(2400\leq2400\), this is valid.

Step6: Check Option F

Substitute \(x = 100\), \(y = 250\) into \(5x + 8y\):
\(5(100)+8(250)=500 + 2000 = 2500\). Since \(2500>2400\), this is invalid.

Wait, earlier for option B: \(8\times300 = 2400\), so \(5(0)+8(300)=2400\), which is equal, so it's valid? But let's re - evaluate option B. The inequality is \(5x + 8y\leq2400\). For \(x = 0\), \(y = 300\), \(5x+8y = 8\times300=2400\), which is equal to 2400, so it satisfies the inequality. But wait, let's check the calculation again for option B:

Wait, \(8\times300 = 2400\), so \(5(0)+8(300)=2400\), which is equal to the right - hand side of the inequality \(2400\). So it should be included. But wait, when we check option B again:

Wait, the original inequality is \(5x + 8y\leq2400\). So if \(5x + 8y = 2400\), it is a solution. So option B: \(x = 0\), \(y = 300\), \(5x + 8y=2400\), so it is a solution.

But wait, let's check all options again:

- Option A: \(x = 480\), \(y = 0\): \(5\times480+8\times0 = 2400\), \(2400\leq2400\): valid.
- Option B: \(x = 0\), \(y = 300\): \(5\times0 + 8\times300=2400\), \(2400\leq2400\): valid.
- Option C: \(x = 200\), \(y = 150\): \(5\times200+8\times150=1000 + 1200 = 2200\leq2400\): valid.
- Option D: \(x = 300\), \(y = 100\): \(5\times300+8\times100=1500 + 800 = 2300\leq2400\): valid.
- Option E: \(x = 400\), \(y = 50\): \(5\times400+8\times50=2000 + 400 = 2400\leq2400\): valid.
- Option F: \(x = 100\), \(y = 250\): \(5\times100+8\times250=500+2000 = 2500>2400\): invalid.

Wait, but this seems to be a mistake. Wait, let's recalculate option B:

\(8\times300 = 2400\), so \(5x + 8y=2400\) when \(x = 0\), \(y = 300\), which is equal to the upper limit, so it is a valid solution.

But let's check the problem again. The inequality is \(5x + 8y\leq2400\). So all the points where \(5x + 8y\) is less than or equal to 2400 are valid.

Now, let's re - check each option:

1. Option A:
\(x = 480\), \(y = 0\)
\(5x+8y=5\times480 + 8\times0=2400+0 = 2400\leq2400\): Valid.

2. Option B:
\(x = 0\), \(y = 300\)
\(5x + 8y=5\times0+8\times300 = 0 + 2400=2400\leq2400\): Valid.

3. Option C:
\(x = 200\), \(y = 150\)
\(5x+8y=5\times200+8\times150 = 1000+1200 = 2200\leq2400\): Valid.

4. Option D:
\(x = 300\), \(y = 100\)
\(5x + 8y=5\times300+8\times100=1500 + 800=2300\leq2400\): Valid.

5. Option E:
\(x = 400\), \(y = 50\)
\(5x+8y=5\times400+8\times50=2000 + 400=2400\leq2400\): Valid.

6. Option F:
\(x = 100\), \(y = 250\)
\(5x + 8y=5\times100+8\times250=500+2000 = 2500>2400\): Invalid.

Answer:

A. 480 chairs and 0 tables, B. 0 chairs and 300 tables, C. 200 chairs and 150 tables, D. 300 chairs and 100 tables, E. 400 chairs and 50 tables