Question

a system of inequalities is graphed on the set of axes below.
which point is a solution to this system?

1. (1,1)
2. (2,-2)
3. (1,8)
4. (4,2)

modeling inequality situations
natasha is planning a school celebration and wants to have live music and food for everyone who attends. she has found a band that will charge her $750 and a caterer who will provide snacks and drinks for $2.25 per person. if her goal is to keep the average cost per person between $2.75 and $3.25, how many people, p, must attend?

1. 225 < p < 325 3) 500 < p < 1000
2. 325 < p < 750 4) 750 < p < 1500

band cost: $750
food cost: $2.25
target cost: $2.75 - $3.25
subtract food cost from target cost – then divide band cost by the difference between the food and target costs.
2.75 - 2.25 = $0.50 now divide 750 by 0.50 = 1500
3.25 - 2.25 = $1.00 now divide 750 by 1.00 = 750
natasha will need between 750 and 1500 people to attend.

Explanation:

First Sub - Question (System of Inequalities Graph)

Step1: Recall Solution Region

A solution to a system of inequalities graphed is a point that lies in the overlapping (shaded) region of all the inequalities.

Step2: Check Each Point

• For point (1,1): Visually, check if it's in the shaded region. From the graph, it seems not in the overlapping shaded area.
• For point (2, - 2): The y - coordinate is negative, and looking at the graph, the shaded region is above the lower line (probably), so this point is not in the solution region.
• For point (1,8): Check the position relative to the lines. The upper and lower bounds of the shaded region don't seem to include this point (y - value is too high relative to the lines).
• For point (4,2): This point lies within the overlapping shaded region of the system of inequalities.

Step1: Define Total Cost

The total cost \(C\) for \(p\) people is the band cost plus the food cost per person times the number of people. So \(C = 750+2.25p\). The average cost per person is \(\frac{750 + 2.25p}{p}=\frac{750}{p}+2.25\).

Step2: Set Up Inequalities

We know that \(2.75<\frac{750}{p}+2.25<3.25\). First, subtract 2.25 from all parts: \(2.75 - 2.25<\frac{750}{p}<3.25 - 2.25\), which simplifies to \(0.5<\frac{750}{p}<1\).

Step3: Solve the Left Inequality

For \(0.5<\frac{750}{p}\), cross - multiply (since \(p>0\), the inequality direction remains the same): \(0.5p < 750\), so \(p < 1500\).

Step4: Solve the Right Inequality

For \(\frac{750}{p}<1\), cross - multiply ( \(p>0\)): \(750 < p\).

Step5: Combine Inequalities

Combining \(p>750\) and \(p < 1500\), we get \(750 < p<1500\).

Answer:

1. (4,2)

Second Sub - Question (Modeling Inequality Situations)