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Use the systems of inequalities graphed below to answer the questions.

What is the inequality symbol of the green line?

What is the inequality symbol of the purple line?

Is the point (-3, -5) a solution to the system of inequalities?

Is the point (1, 6) a solution to the system of inequalities?

3 answers

To determine the inequality symbols for the green and purple lines, we typically follow these rules based on how the lines are graphed:

  1. If the line is solid (includes points on the line), the inequality will be ≤ or ≥.
  2. If the line is dashed (does not include points on the line), the inequality will be < or >.

Answering the Questions:

  1. What is the inequality symbol of the green line?

    • If the green line is solid, the inequality symbol is either ≤ or ≥. If it is dashed, it is either < or >.

  2. What is the inequality symbol of the purple line?

    • If the purple line is solid, it is ≤ or ≥. If it is dashed, it is < or >.

  3. Is the point (-3, -5) a solution to the system of inequalities?

    • Substitute (-3, -5) into each inequality and check if it satisfies all of them. If it does, then it is a solution.

  4. Is the point (1, 6) a solution to the system of inequalities?

    • Substitute (1, 6) into each inequality and check if it satisfies all of them. If it does, then it is a solution.

Since I don't have access to the specific graph or the inequalities themselves, please apply these techniques to your graph to find the correct answers:

  • Identify whether the lines are solid or dashed.
  • Substitute the given points into each inequality to see if those points satisfy the conditions of the inequalities.

If you can provide the specific inequalities or graphs, I can assist you in more detail!

Janelle and her friends are selling bracelets and headbands ahead of the school’s pep rally to support the team during homecoming. They figured they would sell the headbands for $4 and the bracelets for $2. They’d like to make more than $200 and sell no more than 100 items.

Select one of the following combinations that Janelle can sell to make at least $200, but have no more than 100 items sold.

(1 point)
Responses

20 bracelets, and 60 headbands
20 bracelets, and 60 headbands

45 Bracelets and 60 headbands
45 Bracelets and 60 headbands

70 Bracelets and 40 headbands
70 Bracelets and 40 headbands

50 Bracelets and 10 Headbands
50 Bracelets and 10 Headbands

To determine which combination of bracelets and headbands meets the requirements of making more than $200 while selling no more than 100 items, we can use the following variables:

  • Let \( b \) be the number of bracelets sold.
  • Let \( h \) be the number of headbands sold.

According to the problem:

  • The price of bracelets is $2 each, so the total revenue from bracelets is \( 2b \).
  • The price of headbands is $4 each, so the total revenue from headbands is \( 4h \).
  • We want the total revenue to be greater than $200: \[ 2b + 4h > 200 \]
  • The total number of items sold must be no more than 100: \[ b + h \leq 100 \]

Now we can test each option given:

  1. 20 bracelets, and 60 headbands

    • Total items: \( 20 + 60 = 80 \) (within limit)
    • Total revenue: \( 2(20) + 4(60) = 40 + 240 = 280 \) (greater than 200)

  2. 45 bracelets, and 60 headbands

    • Total items: \( 45 + 60 = 105 \) (exceeds limit)
    • No need to check revenue since it exceeds the item limit.

  3. 70 bracelets, and 40 headbands

    • Total items: \( 70 + 40 = 110 \) (exceeds limit)
    • No need to check revenue since it exceeds the item limit.

  4. 50 bracelets, and 10 headbands

    • Total items: \( 50 + 10 = 60 \) (within limit)
    • Total revenue: \( 2(50) + 4(10) = 100 + 40 = 140 \) (not greater than 200)

From this analysis, the only option that meets both conditions (more than $200 in revenue and no more than 100 items sold) is the first option:

20 bracelets, and 60 headbands.