Solving Systems of Inequalities

We need to write a system of inequalities that models Ali's situation. Let's break down what is given.

• Stationery store: $ 8 per hour.
• Post office: $ 10 per hour.
• Workload: at most 25 hours per week.
• Desired income: at least $ 50 per week.

Let's define some variables. Let x be the number of hours Ali works per week at the stationery store and y be the number of hours he works per week at the post office. Since Ali makes $ 8 per hour at the stationery store and $ 10 per hour at the post office, we can write an expression representing his weekly income. Weekly Income 8x + 10y From the given information, we know that Ali wants to earn at least $ 50 per week. With this information we can write the first inequality.

8x + 10y ≥ 50

The number of hours Ali plans to work every week is represented by the expression x+y. To write the second inequality, we will use the fact that Ali decided to work at most 25 hours per week.

x + y ≤ 25

Consequently, we have written a system of inequalities modeling Ali's situation. 8x+10y ≥ 50 & (I) x+y ≤ 25 & (II) Our next step is to solve this system graphically. To do this, we start by writing both inequalities in slope-intercept form.

Let's continue by graphing both inequalities one at a time. First, we write the boundary line. ccc Inequality & & Boundary Line [0.5em] y ≥ -4/5x + 5 & & y = -4/5x + 5 Since the inequality is non-strict, we will draw the boundary line solid. To determine which region of the plane we have to shade, let's test (0,0) into the inequality.

We got a false statement. Therefore, we will shade the region not containing the origin.

In a similar fashion, let's graph the second inequality.

Since the second inequality is also non-strict, we will draw the boundary line solid. Also, testing (0,0) into the inequality produced a true statement, implying that we need to shade the region containing the origin. Let's graph this second region on the same coordinate plane we graphed the first inequality.

Finally, let's remove the non-overlapping parts so that we end only with the solution set. Note that, based on the context, x and y cannot be negative. Therefore, we will also limit the solution set to the first quadrant.

Comparing this solution set to the four answer choices, we conclude that the correct choice is B.

To determine which of the options is the solution set, let's graph the system of inequalities. 3x - 6y > 30 & (I) y ≥ 3x-3 & (II) First, we will write the inequalities in slope-intercept form.

Before plotting the first inequality, we will need to identify the corresponding boundary line. ccc Inequality & & Boundary Line [0.5em] y < 1/2x-5 & & y = 1/2x-5 To determine which region to shade, we will test (0,0) by substituting each coordinate into the inequality.

Since got a false statement, the region to be shaded is the one that does not contain the origin. Additionally, because the inequality is strict, the boundary line is dashed. We are now ready to plot our first inequality.

Next, let's plot the second inequality. Again, we start by writing its boundary line. ccc Inequality & & Boundary Line [0.25em] y ≥ 3x-3 & & y = 3x-3 As we did with the first inequality, to determine which region to shade, let's substitute (0,0) into the inequality.

This time we got a true statement, which implies that the we have to shade the region containing the origin. Since the second inequality is non-strict, the boundary line is solid. Let's graph this inequality on the same coordinate plane we graphed the first inequality.

Finally, let's graph only the solution set of the system of inequalities — the overlapping region.

Comparing the solution set we got with the four given answer choices, we conclude that the correct choice is D.

To verify whether Dominika's graph is correct, let's model the desired situation. From her graph, we can see the use she gave to each variable. x &-- time worked during a shift y &-- dollars earned per shift Dominika earns a fixed amount of $12 per shift. Then, no matter the number of hours, she will earn a minimum of $12. This leads us to set the first inequality. y ≥ 12 However, she can increase her salary in at most $4 per hour. In other words, the maximum amount of money that Dominika can earn in one shift is 4x+12. With this information, we can write a second inequality. Dominika's salary is at most4x+12 ⇓ y ≤ 4x + 12 In a similar fashion, Diego earns a fixed amount of $8 per shift, but also can increase his income with an extra commission that is above $1 per hour. Therefore, we can write two more inequalities. y &≥ 8 [0.1cm] y &> x + 8 Let's form a system with the four inequalities we wrote. y≥ 12 & (I) y≤ 4x+12 & (II) y≥ 8 & (III) y>x+4 & (IV) Note that all the inequalities are written in slope-intercept form. Then, let's proceed to write the boundary lines and to determine which region to shade.

Using the information in the table, let's graph the system of inequalities.

Finally, let's graph only the solution set — the overlapping region.

This region represents the situation in which Dominika and Diego earn the same amount of money in a shift. As we can see, it is a bit different from the one Dominika graphed. Then, we have to conclude that her graph is not correct.

Note that in Dominika's graph, the point (1,10) lies inside the shaded region. This would imply that, on a certain shift, Dominika and Diego worked one hour and earned 10 dollars each but this is not possible because Dominika earns $12 per shift.

To determine whether a system of inequalities with parallel boundaries has no solution, let's consider the simplest pair of parallel lines and different inequalities involving them. For example, we can take two horizontal lines. y=2 and y=-2 These will be our parallel boundaries. Now let's change the equal signs to any inequality symbol to get a system of inequalities. y>2 y< -2 We can graph it on a coordinate plane as shown below.

Since we can see that the graphs of the two inequalities do not intersect, this system of inequalities has no solutions. Is it always like this? What if we try a different system with the same boundary lines? Let's try swapping the inequality symbols. y<2 y> -2 Let's graph it!

As we can see, in this case the solution is the region between the two boundary lines. Thus, we can already conclude that the given claim is sometimes true.

If a system of inequalities has parallel boundaries, it has no solution.