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 35

Chapter 6, Problem 35

Graph the solution set of each system of inequalities or indicate that the system has no solution.
$\begin{cases}\)x $\leq$ 2 $\y$ $\geq$ -1\(\end{cases}$

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Identify the inequalities in the system: $x \leq 5$ and $y \geq -6$.

Graph the boundary line for $x \leq 5$. This is a vertical line at $x = 5$. Since the inequality is $\leq$, shade the region to the left of this line (including the line itself).

Graph the boundary line for $y \geq -6$. This is a horizontal line at $y = -6$. Since the inequality is $\geq$, shade the region above this line (including the line itself).

The solution set to the system is the intersection of the two shaded regions: the area to the left of $x = 5$ and above $y = -6$.

Check a test point in the intersection region (for example, $(0,0)$) to confirm it satisfies both inequalities.

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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 shading the region of the coordinate plane that satisfies the inequality. For example, x ≤ 5 means shading all points to the left of or on the vertical line x = 5. This visual representation helps identify all possible solutions.

Boundary Lines and Their Types

The boundary line of an inequality is the line where the inequality changes from true to false. For '≤' or '≥', the boundary line is solid, indicating points on the line satisfy the inequality. For '<' or '>', the line is dashed, showing points on the line are not included.

Solution Set of a System of Inequalities

The solution set of a system of inequalities is the intersection of the shaded regions for each inequality. It includes all points that satisfy every inequality simultaneously. If no overlap exists, the system has no solution.

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Systems of Inequalities

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. 3x - 2y = − 5 4x + y = 8

Textbook Question

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}\)x^3 + y = 0 $\x$^2 - y = 0\(\end{cases}$

Textbook Question

The perimeter of a rectangle is 26 meters and its area is 40 square meters. Find its dimensions.

Textbook Question

Write the partial fraction decomposition of each rational expression. 6x²-x+1/(x³ + x²+ x +1)

Textbook Question

Write the partial fraction decomposition of each rational expression. $\frac{x^3 + x^2 + 2}{(x^2 + 2)^2}$

Graph the solution set of each system of inequalities or indicate that the system has no solution. {x≤2y≥−1\(\begin{cases}\)x \(\leq\) 2 \(\y\) \(\geq\) -1\(\end{cases}\){x≤2y≥−1

Verified video answer for a similar problem:

Key Concepts

Graphing Linear Inequalities

Boundary Lines and Their Types

Solution Set of a System of Inequalities

Graph the solution set of each system of inequalities or indicate that the system has no solution.
$\begin{cases}\)x \(\leq\) 2 \(\y\) \(\geq\) -1\(\end{cases}$