Question

a landscaping company placed two orders with a nursery but lost their receipts. the first order was for 13 bushes and 4 trees and cost at least $460. the second order was for 6 bushes and 2 trees but cost less than $250. inequalities: what do the variables represent: list 2 possible solutions and their meaning:

Explanation:

Step1: Define variables

Let \(x\) represent the cost of one bush and \(y\) represent the cost of one tree.

Step2: Form first - order inequality

The first order: 13 bushes and 4 trees cost at least \(460\). So, \(13x + 4y\geq460\).

Step3: Form second - order inequality

The second order: 6 bushes and 2 trees cost less than \(250\). So, \(6x + 2y<250\). We can simplify the second inequality by dividing throughout by 2: \(3x + y<125\), then \(y < 125 - 3x\).

Step4: Find possible solutions

Substitute \(y\) in the first inequality.
Substitute \(y=125 - 3x\) (the boundary of the second inequality) into \(13x + 4y\geq460\).
\(13x+4(125 - 3x)\geq460\)
\(13x + 500-12x\geq460\)
\(x+500\geq460\), \(x\geq - 40\). Since \(x>0\) (cost can't be negative).
Let's try \(x = 20\)
If \(x = 20\), from \(y<125 - 3x\), \(y<125-3\times20=125 - 60 = 65\)
Substitute into \(13x + 4y\): \(13\times20+4y=260 + 4y\).
We want \(260+4y\geq460\), \(4y\geq200\), \(y\geq50\)
So one solution is \(x = 20,y = 55\)
Another solution: Let \(x=25\)
From \(y<125 - 3x\), \(y<125-75 = 50\)
Substitute into \(13x + 4y\): \(13\times25+4y=325+4y\)
We want \(325 + 4y\geq460\), \(4y\geq135\), \(y\geq33.75\)
Let \(y = 40\)

Answer:

• Inequalities: \(13x + 4y\geq460\) and \(6x + 2y<250\) (or \(3x + y<125\))
• Variable representation: \(x\) is the cost of one bush, \(y\) is the cost of one tree
• Possible solutions:
• Solution 1: \(x = 20,y = 55\). This means each bush costs \(20\) dollars and each tree costs \(55\) dollars. For the first order (\(13\) bushes and \(4\) trees), the cost is \(13\times20 + 4\times55=260+220 = 480\geq460\). For the second order (\(6\) bushes and \(2\) trees), the cost is \(6\times20+2\times55 = 120 + 110=230<250\)
• Solution 2: \(x = 25,y = 40\). This means each bush costs \(25\) dollars and each tree costs \(40\) dollars. For the first order (\(13\) bushes and \(4\) trees), the cost is \(13\times25+4\times40=325 + 160=485\geq460\). For the second order (\(6\) bushes and \(2\) trees), the cost is \(6\times25+2\times40=150 + 80 = 230<250\)