Step 1 : Recall what an inequality is

In mathematics a **linear inequality** is an inequality which involves a linear function. A linear inequality contains one of the symbols of inequality. It shows the data which is not equal in graph form.

- < is less than
- > is greater than
- ≤ is less than or equal to
- ≥ is greater than or equal to
- ≠ is not equal to
- = is equal to

Step 2: Identify the x- and the y-variables from the key words given and write the first inequality.

Let x = no. of bracelets sold or made

Let y = no. of earrings sold or made

Since the number of bracelets is 2, the number of earrings is 3 we write

2 x + 3 y \geq 50

Step 3: Identify the x variable in the key words given and write the second inequality

Since Sarah knows she will sell more than 50 bracelets we write x >50

where x is the number of bracelets

Step 4: Graph the first inequality

Write the inequality as a linear equation

2 x + 3 y = 500

Recall on how to find the x-intercept

Plug in y = 0 to get x = 250. So our intercept is (250,0)

Recall on how to find the y-intercept

Plug in x = 0 to get y = \frac{500}{3} = 166.666

Join the points to get a straight line. Since the inequality is \geq shade the region above the line.

Step 5: Graph the second inequality x>50

Write the inequality as x = 50

Plot (50,0) on the graph and draw a dotted line through it because the inequality shows >

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Step 5 : Shade the point of intersection

Step 6: Identify a solution

Recall what a solution to an inequality means

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2x + 3y \geq 500

(100,100) is a solution to the inequality since plugging in x = 100 and y = 100 we get 500= 500

(200,100) is a solution to the inequality since plugging in x = 200 and y = 100 we get 400 + 300 = 700 >500

(200,200) is a solution to the inequality since plugging in x = 200 and y = 200 we get 400 + 600 =1000 > 500

Pick any one of these solutions as the answer.