Emma earns $6 each time she mows the lawn and $8 per hour for babysitting. She is saving up to buy a new pair of jeans that cost $48. If she mows the lawn x times and babysits for y hours, which graph shows the amount of work she needs to complete to earn at least enough to purchase the new jeans?

Question

A graph titled Emma apostrophe s Work shows Mowed lawns on the x-axis, from 0 to 10, and Hours of babysitting on the y-axis, from 0 to 50. A solid line has a negative slope and goes through (0, 48) and (8, 0). Everything to the right of the line is shaded.

A graph titled Emma apostrophe s Work shows Mowed lawns on the x-axis, from 0 to 10, and Hours of babysitting on the y-axis, from 0 to 50. A solid line has a negative slope and goes through (0, 48) and (8, 0). Everything to the left of the line is shaded.

A graph titled Emma apostrophe s Work shows Mowed lawns on the x-axis, from 0 to 10, and Hours of babysitting on the y-axis, from 0 to 10. A solid line has a negative slope and goes through (0, 6) and (8, 0). Everything to the right of the line is shaded.

A graph titled Emma apostrophe s Work shows Mowed lawns on the x-axis, from 0 to 10, and Hours of babysitting on the y-axis, from 0 to 10. A solid line has a negative slope and goes through (0, 6) and (8, 0). Everything to the left of the line is shaded.

Mark this and return

Answer 1

To solve the problem, we first need to express the total amount of money Emma earns through mowing lawns and babysitting in relation to her goal of earning at least $48.

Emma earns $6 for each lawn mowed (x times) and $8 for each hour babysat (y hours). Therefore, her total earnings can be expressed as:

\[ 6x + 8y \]

We need this to be at least $48:

\[ 6x + 8y \geq 48 \]

To find the boundary line of this inequality, we can rewrite it as:

\[ 8y = 48 - 6x \] \[ y = 6 - \frac{3}{4}x \]

This equation has a y-intercept of 6 (where x = 0) and an x-intercept of 8 (where y = 0). The line will have a negative slope because as x increases (more lawns mowed), y must decrease (less time babysitting) to keep the total earnings at or above $48.

Now, let's analyze how to shade the graph based on the inequality \( 6x + 8y \geq 48 \). We are looking for the region where Emma's earnings meet or exceed $48, which will be above or on the line.

Among the options presented:

1. A graph with a solid line through (0, 48) and (8, 0), with shading to the right indicates that the earnings are increasing with more work.
2. A graph with a solid line through (0, 48) and (8, 0), with shading to the left, suggests fewer earnings, which is incorrect.
3. A graph with a line through (0, 6) and (8, 0) with shading to the right doesn't match our calculated intercepts (this is not the correct line).
4. A graph with a line through (0, 6) and (8, 0) with shading to the left is also incorrect.

The correct graph is the first one: it shows a solid line with the correct intercepts (0, 6) and (8, 0) with shading to the right, representing all the combinations of mowing lawns and babysitting hours that allow Emma to earn at least $48.

So the answer is the first graph.