Answer:
Here is how to write and graph an inequality for the budget problem:
Let x be the number of large boxes and y be the number of small boxes. The total cost of the boxes must be less than or equal to the budget of $500. So, we can write the inequality as:
65x + 35y ≤ 500
To graph this inequality, we need to find the boundary line and the shaded region. The boundary line is the equation that we get when we replace the inequality sign with an equal sign, which is:
65x + 35y = 500
To graph this line, we need to find two points on it. We can do this by plugging in some values for x or y and solving for the other variable. For example, if x = 0, then y = 500/35 = 14.29. If y = 0, then x = 500/65 = 7.69. So, two points on the line are (0, 14.29) and (7.69, 0). We can plot these points on a coordinate plane and draw a solid line through them, since the inequality includes the equal sign.
To find the shaded region, we need to test a point that is not on the line and see if it satisfies the inequality. For example, we can test the point (0, 0), which is the origin. Plugging in x = 0 and y = 0 into the inequality, we get:
65(0) + 35(0) ≤ 500
0 ≤ 500
This is true, so the point (0, 0) is a solution to the inequality. This means that the shaded region is the area that includes the point (0, 0) and is below the line.
If Renaldo buys 6 small gift boxes, then y = 6. To find how many large gift boxes he can afford, we need to plug in y = 6 into the inequality and solve for x. We get:
65x + 35(6) ≤ 500
65x + 210 ≤ 500
65x ≤ 290
x ≤ 290/65
x ≤ 4.46
This means that Renaldo can buy at most 4 large gift boxes if he buys 6 small gift boxes. However, since x has to be a whole number, the largest possible value for x is 4. So, Renaldo can buy 4 large gift boxes and 6 small gift boxes and still stay within his budget.
I hope this helps!
Explanation: