Two-Variable Inequalities

In elementary algebra, we have learned how to solve systems of equations. The solution to a system of linear equations is the point where the graphs of the lines intersect. The solution to a system of linear inequalities is every point in a region of the graph where the inequalities overlap, rather than the point of intersection of the lines (Slavin, 2001).

This week's assignment required to solve problem 68 on page 539 (Dugopolski, 2012). I will be giving a detailed presentation on math required for the solution to this problem; the accompanying graph shows all of the possibilities for the number of refrigerators and the number of TVs that will fit into an 18-wheeler. The point-slope form of a linear equation to write the equation itself can now be used. These are the steps we take to solve our linear inequality. I will start with the point-slope form. Substitute slope form with (300, 0) for the x and y. Next, we are going to use the distributive property and then add 330 to both sides and divided both sides by -3 and cancel out like terms.

The graph has a solid line rather than a dashed line indicating that points on the line itself are part of the solution set. This will be true anytime the inequality symbol has equal to the bar.

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a) Write an inequality to describe this region.

p = y1-y2 /x1-x2 = 330 – 0 / 0-110 = -3/1 the slope is -3/1 or -3 y – y1 = p(x – x1) y– 330 = - 3 / 1(x-0) y= - 3x/1 + 330 -3x/1 +330 = y expression switch by place the y on the right hand side -3x/-3 = y/-3 – 330/ -3 divide each equation by -3 and cancel out like terms -3y = 1x + 110 -3y + 1x < 110.

b) Will the truck hold 71 refrigerators and 118 TVs? In this problem I will be substituting 71 where the y is for refrigerators and 118 where the x is for TVs to determine if the truck will hold them.

-3 (71) + 1 (118) < 110 -213 + 118 < 110 -95 < 110 Which means that the truck will not hold 71 refrigerators and 118 TVs.

c) Will the truck hold 51 refrigerators and 176 TVs?

This problem is similar to the previous one.

-3 (51) + 1 (176) < 110 -153 +176 < 110 23 < 110 yes, the truck will hold at least 51 refrigerators and 176 TVs.

The Burbank Buy More store is going to make an order which will include at most 60 refrigerators. What is the maximum number of TVs which could also be delivered on the same 18-wheeler? Describe the restrictions this would add to the existing graph. Solving for y:

1(60) + -3y < 110 -3y < -60 + 110 add 110 to -60 to get 50 -3y < 50 divide both terms by -3 -3y/-3 > 50/-3 signs flip y > -50/3 or y = 16.

There will be no added restriction because the maximum numbers of TVs The next day, the Burbank Buy More decides they will have a television sale so they change their order to include at least 200 TVs. What is the maximum number of refrigerators which could also be delivered in the same truck? Describe the restrictions this would add to the original graph. 1x + -3 (200) < 110

x < 600 + 100 x = < 710 If 200 TVs are ship.

When graphing linear inequalities and they are greater than or less than you will use a dashed line. When the inequality is greater than or equal to or less than or equal to you will then use a solid line, which indicates that the points on the line are part of the solution set (Slavin, 2001). In this paper, I have shown you a linear equation and broke down each step in solving the equation. I used the method of substitution for the variables and explained what the equations represent.

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