Ch. 5 - Systems of Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

Problem 61

Chapter 6, Problem 61

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. 3x+y≤6, 2x−y≤−1, x>−2, y<4

Verified step by step guidance

Step 1: Rewrite each inequality in a form that is easier to graph. For example, express each inequality in terms of y when possible. For the first inequality, 3x + y ≤ 6, solve for y to get y ≤ -3x + 6.

Step 2: For the second inequality, 2x - y ≤ -1, solve for y to get y ≥ 2x + 1 (note the inequality direction changes when subtracting y and rearranging).

Step 3: The third inequality is x > -2, which represents all points to the right of the vertical line x = -2 (not including the line itself).

Step 4: The fourth inequality is y < 4, which represents all points below the horizontal line y = 4 (not including the line itself).

Step 5: Graph each boundary line on the coordinate plane: y = -3x + 6 (solid line because of ≤), y = 2x + 1 (solid line because of ≥), x = -2 (dashed line because of >), and y = 4 (dashed line because of <). Then shade the regions that satisfy each inequality and find the overlapping region that satisfies all inequalities simultaneously.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Graphing Linear Inequalities

Graphing linear inequalities involves plotting the boundary line of the inequality and shading the region that satisfies the inequality. The boundary line is solid if the inequality includes equality (≤ or ≥) and dashed if it does not (< or >). This visual representation helps identify all possible solutions to the inequality.

System of Inequalities

A system of inequalities consists of two or more inequalities considered simultaneously. The solution set is the intersection of the regions that satisfy each inequality individually. Understanding how to find this common region is essential for solving and graphing the system.

Feasibility and No Solution Cases

Some systems of inequalities have no solution if their solution regions do not overlap. Recognizing when the shaded regions do not intersect is crucial to conclude that the system is inconsistent. This concept helps avoid incorrect assumptions about solution sets.

Recommended video:

Guided course

6:19

Systems of Inequalities

Related Practice

Textbook Question

Find the partial fraction decomposition of 4x²+5x-9/(x³- 6x-9)

views

Textbook Question

Write the given sentences as a system of inequalities in two variables. Then graph the system. The sum of the x-variable and the y-variable is at most 4. The y-variable added to the product of 3 and the x-variable does not exceed 6.

views

Textbook Question

In Exercises 63–64, write each sentence as an inequality in two variables. Then graph the inequality. The y-variable is at least 4 more than the product of -2 and the x-variable.

Textbook Question

Graph the solution set of each system of inequalities or indicate that the system has no solution.

$\begin{cases}\)x $\geq$ 0 $\y$ $\geq$ 0 \\2x + 5y < 10 \\3x + 4y $\leq$ 12\(\end{cases}$

Textbook Question

In Exercises 57–59, graph the region determined by the constraints. Then find the maximum value of the given objective function, subject to the constraints. This is a piecewise function. Refer to the textbook.

views

Textbook Question

Exercises 57–59 will help you prepare for the material covered in the next section. Solve: $\begin{cases}\)A + B = 3 \\2A - 2B + C = 17 \\4A - 2C = 14\(\end{cases}$