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Chapter 5: Problem 49

Graph the solution set. If there is no solution, indicate that the solution set is the empty set. \(2 x+5 y \leq 5\) \(-3 x+4 y \geq 4\)

Short Answer

Expert verified
Shade the region where the two shaded areas overlap on the graph.

Step by step solution

01

Draw the Boundary Lines

Start by converting the inequalities into equations. For the first inequality, convert it to the equation: \(2x + 5y = 5\). For the second inequality, convert it to the equation: \(-3x + 4y = 4\). Graph these lines on the coordinate plane.
02

Determine the Border Line Points

Find the intercepts of the boundary lines by setting \(x\) and \(y\) to 0 alternately. For \(2x + 5y = 5\), set \(x = 0\) to find \(y\)-intercept: \(0 + 5y = 5\) implies \(y = 1\).Set \(y = 0\) to find \(x\)-intercept: \(2x + 0 = 5\) implies \(x = 2.5\). For \(-3x + 4y = 4\), set \(x = 0\): \(4y = 4\) implies \(y = 1\).Set \(y = 0\): \(-3x = 4\) implies \(x = -\frac{4}{3}\).
03

Graph the Boundary Lines

Using the points found in Step 2, draw the lines \(2x + 5y = 5\) and \(-3x + 4y = 4\) on the coordinate plane. Use a solid line because the inequalities include equality (鈮� and 鈮�).
04

Determine the Shaded Area for Each Inequality

For \(2x + 5y \leq 5\), test a point not on the line (0,0) for convenience: \(2(0) + 5(0) \leq 5\) which is true, so shade the side including the origin. For \(-3x + 4y \geq 4\), test the point (0,0): \(-3(0) + 4(0) \geq 4\) which is false, so shade the opposite side of the origin.
05

Find the Intersection of the Shaded Regions

Determine where the shaded regions of both inequalities overlap. This is the solution set of the system. If no such overlap exists, the solution set is empty.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Boundary Lines
Boundary lines are crucial when graphing linear inequalities. These lines demarcate the regions of the coordinate plane where the inequality might hold true. To begin, we convert inequalities into equalities. For instance, convert the inequality \(2x + 5y \leq 5\) into the equation \(2x + 5y = 5\). Similarly, \(-3x + 4y \geq 4\) becomes \(-3x + 4y = 4\). Plot these lines accurately on the coordinate plane. The points where these lines cross the x-axis and y-axis help in plotting the boundary lines correctly.
Inequality Solutions
An inequality solution tells us where the solutions to an inequality lie on the coordinate plane. After drawing the boundary lines, we decide which side of the line to shade. For example, take the inequality \(2x + 5y \leq 5\). We choose a test point not on the line to check if the inequality holds. Testing (0,0): \(2(0) + 5(0) \leq 5\). This is true, so we shade this region. Similarly, for \(-3x + 4y \geq 4\), test point (0,0) gives: \(-3(0) + 4(0) \geq 4\), which is false; thus, we shade the opposite side. This helps in understanding where the true solutions lie.
Coordinate Plane Intersections
Intersections on a coordinate plane are pivotal for solving systems of inequalities. They indicate where boundary lines cross and help in visualizing how inequalities interact. For our lines \(2x + 5y = 5\) and \(-3x + 4y = 4\), plotting these lines enables us to easily spot their intersection points. These intersections are critical because they can signify boundaries within which solutions might exist. By thoroughly analyzing these intersections, it becomes clear how individual inequalities overlap and form a collective solution set.
Shaded Regions
Shaded regions on a coordinate plane indicate where the solutions for an inequality lie. For each inequality, shading helps visualize feasible solution areas. Once the boundary lines are drawn, the solution region is determined by shading. Let's consider the example again. For \(2x + 5y \leq 5\), shading the area that includes the origin as our test point (0,0) worked. For \(-3x + 4y \geq 4\), shading the region away from the origin was correct. Where these shaded areas converge indicates the solution set of both inequalities. If the shaded regions don't overlap, then there is no common solution.

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Most popular questions from this chapter

Solve the system of equations by using the addition method. (See Examples \(3-4)\) $$ \begin{array}{l} 2 x+11 y=4 \\ 3 x-6 y=5 \end{array} $$

Solve the system using any method. $$ \begin{array}{l} 2 x-7 y=2400 \\ -4 x+1800=y \end{array} $$

A pharmacist wants to mix a \(30 \%\) saline solution with a \(10 \%\) saline solution to get \(200 \mathrm{~mL}\) of a \(12 \%\) saline solution. How much of each solution should she use?

Josh makes \(\$ 24 /\) hr tutoring chemistry and \(\$ 20 / \mathrm{hr}\) tutoring math. Let \(x\) represent the number of hours per week he spends tutoring chemistry. Let \(y\) represent the number of hours per week he spends tutoring math. a. Write an objective function representing his weekly income for tutoring \(x\) hours of chemistry and \(y\) hours of math. b. The time that Josh devotes to tutoring is limited by the following constraints. Write a system of inequalities representing the constraints. \- The number of hours spent tutoring each subject cannot be negative. \- Due to the academic demands of his own classes he tutors at most \(18 \mathrm{hr}\) per week. \- The tutoring center requires that he tutors math at least 4 hr per week. \- The demand for math tutors is greater than the demand for chemistry tutors. Therefore, the number of hours he spends tutoring math must be at least twice the number of hours he spends tutoring chemistry. c. Graph the system of inequalities represented by the constraints. d. Find the vertices of the feasible region. e. Test the objective function at each vertex. f. How many hours tutoring math and how many hours tutoring chemistry should Josh work to maximize his income? g. What is the maximum income? h. Explain why Josh's maximum income is found at a point on the line \(x+y=18\).

Solve the system of equations by using the substitution method. (See Example 2\()\) $$ \begin{array}{l} 3 x=2 y-11 \\ 6+5 x=1 \end{array} $$
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