Question

inequalities

1. a jewelry store employee needs to decide how many pieces of jewelry to put on display. the store manager wants the number of necklaces to be at least five more than the number of rings. the store employee represents this restriction as ( y geq 5 + x ), where ( x ) is the number of rings and ( y ) is the number of necklaces. which ordered pairs represent viable solutions in terms of context? select all that apply.

a) ( (-1, 6) )

b) ( (4, 8) )

c) ( (9, 4) )

d) ( (3, 10) )

e) ( (5, 11.5) )

Explanation:

Step1: Analyze the context

The number of rings (\(x\)) and necklaces (\(y\)) must be non - negative integers because we can't have a negative number of jewelry pieces and we can't have a fraction of a jewelry piece (in the context of counting whole pieces). Also, we need to check if \(y\geq5 + x\).

Step2: Check option a \((- 1,6)\)

The number of rings \(x=-1\) is negative. Since the number of rings can't be negative, this ordered pair is not a viable solution.

Step3: Check option b \((4,8)\)

First, \(x = 4\) (non - negative integer) and \(y = 8\) (non - negative integer). Now check the inequality \(y\geq5 + x\). Substitute \(x = 4\) into \(5 + x\), we get \(5+4=9\). But \(8<9\), so \(y = 8\) does not satisfy \(y\geq5 + x\). Wait, there is a mistake here. Wait, \(5 + x=5 + 4 = 9\), and \(y = 8\), \(8<9\), so this is wrong. Wait, maybe I miscalculated. Wait, \(x = 4\), \(y = 8\). The inequality is \(y\geq x + 5\), so \(8\geq4 + 5=9\)? No, \(8<9\). So this is not a solution. Wait, maybe the option d is \((3,10)\) (I think there is a typo, it should be \((3,10)\)).

Step4: Check option c \((9,4)\)

\(x = 9\) (non - negative integer), \(y = 4\) (non - negative integer). Check \(y\geq5 + x\). \(5+x=5 + 9=14\), and \(4<14\), so \(y = 4\) does not satisfy \(y\geq5 + x\). Also, \(y = 4<x = 9\), and the number of necklaces should be at least 5 more than the number of rings, so this is not a solution.

Step5: Check option d \((3,10)\) (assuming the typo is fixed, it should be \((3,10)\))

\(x = 3\) (non - negative integer), \(y = 10\) (non - negative integer). Check \(y\geq5 + x\). \(5+x=5 + 3=8\). Since \(10\geq8\), and \(x = 3\), \(y = 10\) are non - negative integers, this ordered pair is a viable solution.

Step6: Check option e \((5,11.5)\)

The number of necklaces \(y = 11.5\) is a non - integer. Since we can't have a fraction of a necklace (in the context of counting whole jewelry pieces), this ordered pair is not a viable solution. Wait, wait, maybe I made a mistake with option b. Wait, let's re - check option b: \(x = 4\), \(y = 8\). \(5+x=9\), \(y = 8\), \(8<9\), so not valid. Wait, maybe the original problem has a typo in option d, maybe it's \((3,10)\). Wait, let's re - evaluate:

Wait, the inequality is \(y\geq x + 5\).

For option b: \(x = 4\), \(y = 8\). \(x+5=9\), \(8<9\)→ not valid.

For option d: \(x = 3\), \(y = 10\). \(x + 5=8\), \(10\geq8\)→ valid. Also, \(x = 3\) and \(y = 10\) are non - negative integers.

Wait, maybe I misread option b. Wait, maybe the option d is \((3,10)\) (with a comma). Let's check again:

Wait, the problem says "d) (3 10)" which is probably a typo for \((3,10)\).

Now, let's re - check all options:

- Option a: \(x=-1\) (invalid, negative)
- Option b: \(x = 4\), \(y = 8\). \(y\geq x + 5\)→ \(8\geq4 + 5=9\)? No.
- Option c: \(x = 9\), \(y = 4\). \(y\geq x + 5\)→ \(4\geq9 + 5=14\)? No.
- Option d: \(x = 3\), \(y = 10\). \(y\geq x + 5\)→ \(10\geq3+5 = 8\). Yes. And \(x = 3\), \(y = 10\) are non - negative integers.
- Option e: \(y = 11.5\) (non - integer, invalid)

Wait, maybe I made a mistake with option b. Wait, \(x = 4\), \(y = 8\). \(x+5=9\), \(y = 8\), \(8<9\), so not valid. So the only valid one is d? But that seems odd. Wait, maybe the original problem has a different set of options. Wait, maybe the user made a typo in option d, maybe it's \((3,10)\) (with a comma). Also, maybe I made a mistake in option b. Wait, \(x = 4\), \(y = 8\). The inequality is \(y\geq x + 5\), so \(8\geq4 + 5\)→ \(8\geq9\) is false. So option d is the only one? Wait, no, maybe I misread the options. Wait, option d is \((3 10)\), probably \((3,10)\). Let's confirm:

- For a: \(x=-1\) (invalid)
- For b: \(x = 4\), \(y = 8\), \(8\geq4 + 5=9\)? No.
- For c: \(x = 9\), \(y = 4\), \(4\geq9+5 = 14\)? No.
- For d: \(x = 3\), \(y = 10\), \(10\geq3 + 5=8\). Yes. And \(x,y\) are non - negative integers.
- For e: \(y = 11.5\) (invalid)

So the viable solution is d) \((3,10)\) (assuming the typo is fixed).

Answer:

d) \((3,10)\) (assuming the typo in the option, it should be \((3,10)\) instead of \((3 10)\))