Exercise 34 Page 254

To solve this exercise, first we have to write the inequality representing the problem. Next we will graph it. Finally, from the graph we will be able to identify and interpret two solutions of the inequality.

Writing the Inequality

We are told that the drama club must sell at least $1500 worth of tickets to cover the expenses of producing the play. This means that the income from selling the tickets must be greater than or equal to $1500. lIncome from selling the tickets ≥ $ 1500 Now we must rewrite the left-hand side of the inequality using variables. Let x be the number of adult tickets and y the number of student tickets. We know that one adult ticket costs $10 and one student ticket costs $6. Using this information, we can write an expression to represent the income. lIncome from selling the tickets = x* $ 10 + y* $ 6 Let's substitute the above expression into our original inequality. x* $ 10 +y* $ 6≥ $ 1500 ⇔ 10x+6y≥ 1500

Graphing the Inequality

Next we will graph the obtained inequality. 10x+6y≥ 1500 To do so we have to draw the boundary line. The equation of a boundary line is written by replacing the inequality symbol with an equals sign. Inequality & Boundary Line 10x+6y ≥ 1500 & 10x+6y = 1500 To draw this line we will first rewrite the equation in slope-intercept form.

Now that the equation is in slope-intercept form, we can identify the slope m and y-intercept b. y=-5/3x+ 250 We will plot the y-intercept and then use the slope to plot another point on the line. The boundary line will be solid because the inequality is not strict. Since x and y both represent the number of tickets, we need to restrict the graph to non-negative values of x and y. Negative values do not make sense in this real-life context.

To decide which side of the boundary line to shade, we will substitute a test point that is not on the boundary line into the given inequality. If the substitution creates a true statement, we shade the region that includes the test point. Otherwise, we shade the opposite region. Let's use ( 0, 0) as our test point.

Since the substitution of the test point did not create a true statement, we will shade the region that does not contain the point.

Identifying and Interpreting the Solutions

Finally, we can identify two solutions of the inequality. A solution of the graphed inequality is a point that lies in the shaded region. Note that since the boundary line is solid, every point lying on it is a solution of the inequality as well. Let's choose our two points!

Both (30,200) and (100,140) are solutions of the inequality. The solution ( 30, 200) means that the drama club sells 30 adult tickets and 200 student tickets. Similarly, (100,140) means that the club sells 100 adult tickets and 140 student tickets.