We are given information about the number of hours we can work at two different jobs. Let's start by organizing the information into inequalities. First, let's assign variables to the hours worked at the different jobs.

Hours worked mowing lawns: $x$
Hours worked at the clothing store: $y$

We are given the rate of pay for each job,

$ 12

an hour for mowing lawns and

$ 10

an hour at the clothing store. Also, we need to make at least

$ 350

per week. We can rewrite this using an inequality.

12 x + 10 y \geq 350

Next, we know that we can work no more than

35

hours in the week. Let's rewrite this as an inequality.

x + y \leq 35

The last bit of information is that we need to work at least

10

hours at the clothing store. We will rewrite this as an inequality as well.

y \geq 10

We now have three inequalities that we will need to graph.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 12 x + 10 y \geq 350 x + y \leq 35 y \geq 10 (I) (II) (III)

Let's start with writing the boundary line for the first equation. We need to isolate the

y -

variable.

12 x + 10 y = 350

SubEqn

$LHS - 12 x = RHS - 12 x$

10 y = 350 - 12 x

DivEqn

$LHS / 10 = RHS / 10$

y = 35 - 1.2 x

CommutativePropAdd

Commutative Property of Addition

y = - 1.2 x + 35

We can now graph the boundary line. Since the inequality is non-strict, we will use a solid line. Also, since it is impossible to work a negative number of hours, we will only consider positive values of

x

and

y .

Now, we can determine where to shade using a test point. We will use the point

(0, 0)

for simplicity. Let's substitute the values into the original inequality.

12 x + 10 y \geq 350

SubstituteII

$x = 0$ , $y = 0$

12 (0) + 10 (0) \geq ? 350

ZeroPropMult

Zero Property of Multiplication

0 ≱ 350 \times

Since

(0, 0)

made the inequality false, we will shade the region that does not contain the point.

We will follow the same process to graph the other two inequalities. To graph the second inequality,

x + y \leq 35,

we need to isolate the

y -

variable by subtracting

x

from both sides.

x + y = 35 \Leftrightarrow y = - x + 35

Since the inequality is non-strict, the line will be solid. We will use the same test point to determine which region we will shade.

x + y \leq 35

SubstituteII

$x = 0$ , $y = 0$

0 + 0 \leq ? 35

AddTerms

Add terms

0 \leq 35 ✓

Since we arrived at a true statement, we will shade the region that does contain the point.

Finally, we can graph the third inequality,

y \geq 10 .

Since the inequality only has a

y -

variable, we know that the boundary line will be horizontal. We will use the same test point to determine where we will shade.

y \geq 10

Substitute

$y = 0$

0 ≱ 10 \times

Since

(0, 0)

made the inequality false, we will shade the region that does not contain the test point.

The solution set is where the shading of all three of the inequalities overlap.

Each point that is in the shaded region represents the possible hours that can be worked at each job. For example, the point $(13, 21)$ indicates that we can work $13$ hours mowing lawns and $21$ hours at the clothing store. This is just one of the possible solutions.

Exercise

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