Question

a cleaning service offers its customers a choice between two plans.

• plan a costs $3,000 and includes 1 year fan unlimited number of cleaning.
• plan b costs $75 per cleaning.

harry wants to choose the less expensive plan. he uses the inequality 3,000 < 75c to decide which plan to choose based on the number of cleaning (c) he expects to need. based on the solution of the inequality, which statement about harry’s choice of plan is true?
a. harry should choose plan a only if he expects to need fewer than 40 cleanings in 1 year.
b. harry should choose plan a only if he expects to need more than 40 cleanings in 1 year.
c. harry can choose either plan and pay the same amount if he expects to need fewer than 40 cleanings in 1 year.
d. harry can choose either plan and pay the same amount if he expects to need more than 40 cleanings in 1 year.

1. pamela keeps apples in 3 bins and 2 crates in her store. each bin can hold no more than 200 pounds. each crate can hold no more than 50 pounds. which graph represents all of the possible weights, in pounds, of apples pamela can keep in her store?
2. hydrogen is a gas at certain temperatures under normal conditions. these temperatures (t), in degrees celsius (°c), are represented by the inequality t ≥ -196. which best describes the meaning of the inequality?

a. hydrogen is a gas if the temperature is below -196°c.
b. hydrogen is a gas only if the temperature is above -196°c.
c. hydrogen is a gas only if the temperature is at or below -196°c.
d. hydrogen is a gas only if the temperature is at or above -196°c.

1. jamal decided that he will sell his stocks if their value per share (x) goes below $5 or above $15. which compound inequality represents the values at which jamal will sell his stocks?

a. x > $5 or x < $15
c. x < $5 or x > $15
b. x > $5 and x < $15
d. x < $5 and x > $15

Explanation:

Step1: Solve the inequality \(3000 < 75c\)

Divide both sides of the inequality by \(75\).
$$\frac{3000}{75}<\frac{75c}{75}$$
$$40 < c$$
This means when \(c>40\), Plan A (\(3000\) dollars) is cheaper than Plan B (\(75c\) dollars). When \(c < 40\), Plan B is cheaper. When \(c = 40\), \(3000=75\times40\), the two plans cost the same.

Step2: Analyze each option

• Option A: If \(c<40\), Plan B is cheaper. So this option is wrong.
• Option B: If \(c > 40\), Plan A is cheaper. But it's not "only if", when \(c = 40\) the costs are equal. So this option is wrong.
• Option C: When \(c<40\), Plan B is cheaper. So this option is wrong.
• Option D: When \(c>40\), Plan A is cheaper. When \(c = 40\), \(3000 = 75\times40\), the two plans cost the same. So this option is correct.

For the hydrogen temperature problem (\(t\geq - 196\)):

Step1: Analyze the inequality \(t\geq - 196\)

The inequality \(t\geq - 196\) means \(t=-196\) or \(t>-196\).

Step2: Analyze each option

• Option A: \(t < - 196\) does not satisfy \(t\geq - 196\). So this option is wrong.
• Option B: \(t>-196\) is part of \(t\geq - 196\), but \(t=-196\) also makes hydrogen a gas. So this option is wrong.
• Option C: \(t\leq - 196\) does not satisfy \(t\geq - 196\). So this option is wrong.
• Option D: \(t\geq - 196\) means \(t=-196\) or \(t > - 196\). So this option is correct.

For Jamal's stock - selling problem:

Step1: Translate the word - problem into an inequality

"Value per share \((x)\) goes below \(\$5\)" is \(x < 5\), "above \(\$15\)" is \(x>15\). Since it's "below \(\$5\) or above \(\$15\)", we use the "or" connective.

Step2: Analyze each option

• Option A: \(x > 5\) or \(x < 15\) does not represent the correct range. So this option is wrong.
• Option B: \(x > 5\) and \(x < 15\) represents the values between \(5\) and \(15\). So this option is wrong.
• Option C: \(x < 5\) or \(x > 15\) correctly represents "below \(\$5\) or above \(\$15\)". So this option is correct.
• Option D: \(x < 5\) and \(x > 15\) has no solution (a number can't be both less than \(5\) and greater than \(15\)). So this option is wrong.

Answer:

For the first problem (cleaning plans): D. Harry can choose either plan and pay the same amount if he expects to need more than 40 cleanings in 1 year.
For the hydrogen temperature problem: D. Hydrogen is a gas only if the temperature is at or above \(-196^{\circ}C\).
For Jamal's stock - selling problem: C. \(x < \$5\) or \(x > \$15\)