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 31

Chapter 6, Problem 31

Graph the solution set of each system of inequalities or indicate that the system has no solution.
$\begin{cases}\)y > 2x - 3 $\y$ < -x + 6\(\end{cases}$

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Identify the two inequalities in the system: $y > 4x - 2$ and $y < -3x + 9$.

Graph the boundary lines for each inequality: the line $y = 4x - 2$ and the line $y = -3x + 9$. Use a solid line if the inequality includes equality (≥ or ≤), but here use dashed lines because the inequalities are strict ( > and < ).

Determine the shading for each inequality: For $y > 4x - 2$, shade the region above the line $y = 4x - 2$. For $y < -3x + 9$, shade the region below the line $y = -3x + 9$.

Find the intersection of the two shaded regions. This overlapping area represents the solution set to the system of inequalities.

If the shaded regions do not overlap, then the system has no solution. Otherwise, the solution set is the region where both inequalities are true simultaneously.

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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 given by the corresponding linear equation and then shading the region that satisfies the inequality. For strict inequalities (>, <), the boundary line is dashed to indicate points on the line are not included. The shaded area represents all solutions that make the inequality true.

System of Inequalities

A system of inequalities consists of two or more inequalities considered simultaneously. The solution set is the intersection of the individual solution regions, meaning only points that satisfy all inequalities are included. If no common region exists, the system has no solution.

Slope-Intercept Form and Interpretation

The slope-intercept form y = mx + b expresses a line where m is the slope and b is the y-intercept. Understanding the slope helps determine the line's direction, while the intercept shows where it crosses the y-axis. This form is essential for quickly graphing the boundary lines of inequalities.

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Slope-Intercept Form

Related Practice

Textbook Question

In Exercises 31–42, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. x = 9-2y x + 2y = 13

Textbook Question

Write the partial fraction decomposition of each rational expression. 5x²+6x+3/(x + 1)(x² + 2x + 2)

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Textbook Question

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}\)2x^2 + y^2 = 18 $\xy$ = 4\(\end{cases}$

Textbook Question

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}\)3x^2 + 4y^2 = 16 \\2x^2 - 3y^2 = 5\(\end{cases}$

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Textbook Question

In Exercises 19–30, solve each system by the addition method. 3x = 4y + 1 3y = 1 - 4x

Textbook Question

Write the partial fraction decomposition of each rational expression. 5x² -6x+7/(x − 1) (x² + 1)

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Graph the solution set of each system of inequalities or indicate that the system has no solution.{y>2x−3y<−x+6\(\begin{cases}\)y > 2x - 3 \(\y\) < -x + 6\(\end{cases}\){y>2x−3y<−x+6

Verified video answer for a similar problem:

Key Concepts

Graphing Linear Inequalities

System of Inequalities

Slope-Intercept Form and Interpretation

Graph the solution set of each system of inequalities or indicate that the system has no solution.
$\begin{cases}\)y > 2x - 3 \(\y\) < -x + 6\(\end{cases}$