Questions: Gabriella is working two summer jobs, making 16 per hour lifeguarding and making 6 per hour walking dogs. In a given week, she can work no more than 13 total hours and must earn a minimum of 120. If x represents the number of hours lifeguarding and y represents the number of hours walking dogs, write and solve a system of inequalities graphically and determine one possible solution. Inequality 1: y ≥ Inequality 2: y ≥ Gabriella could work hours lifeguarding and hours walking dogs.

Gabriella is working two summer jobs, making 16 per hour lifeguarding and making 6 per hour walking dogs. In a given week, she can work no more than 13 total hours and must earn a minimum of 120. If x represents the number of hours lifeguarding and y represents the number of hours walking dogs, write and solve a system of inequalities graphically and determine one possible solution.

Inequality 1: y ≥

Inequality 2: y ≥

Gabriella could work hours lifeguarding and hours walking dogs.

Transcript text: Gabriella is working two summer jobs, making $\$ 16$ per hour lifeguarding and making $\$ 6$ per hour walking dogs. In a given week, she can work no more than 13 total hours and must earn a minimum of $\$ 120$. If $x$ represents the number of hours lifeguarding and $y$ represents the number of hours walking dogs, write and solve a system of inequalities graphically and determine one possible solution. Inequality 1: $y \geq$ $\square$ Inequality 2: $y \geq$ $\square$ Gabriella could work $\square$ hours lifeguarding and $\square$ hours walking dogs.

Solution

Solution Steps

Step 1: Define the variables and write the inequalities.

Let _x_ be the number of hours lifeguarding and _y_ be the number of hours walking dogs.

Inequality 1 (Total hours): x + y ≤ 13

Inequality 2 (Minimum earnings): 16x + 6y ≥ 120

Step 2: Rewrite the inequalities in slope-intercept form.

Inequality 1: y ≤ -x + 13

Inequality 2: y ≥ (-16/6)x + 20, which simplifies to y ≥ (-8/3)x + 20

Step 3: Graph the inequalities.

Graph the line y = -x + 13. Since the inequality is less than or equal to, shade the region below the line.

Graph the line y = (-8/3)x + 20. Since the inequality is greater than or equal to, shade the region above the line.

Step 4: Identify the overlapping region.

The overlapping shaded region represents the feasible solutions.

Step 5: Find a possible solution.

One point in the overlapping region is (6, 4). This represents 6 hours lifeguarding and 4 hours walking dogs. Another point is (5,5). This represents 5 hours lifeguarding and 5 hours dog walking. Many other feasible points exist in the shaded overlap region.

Final Answer

Gabriella could work 6 hours lifeguarding and 4 hours walking dogs, or 5 hours lifeguarding and 5 hours walking dogs. Other solutions are possible within the overlapping shaded regions on the graph.

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