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 10

Chapter 6, Problem 10

Graph each inequality. x≤−3

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Identify the inequality given: $x \leq -3$. This means we are looking for all values of $x$ that are less than or equal to $-3$.

Draw a number line and locate the point $-3$ on it. This point will be important as it is the boundary of the inequality.

Since the inequality includes $\leq$ (less than or equal to), use a solid circle or dot at $-3$ to indicate that $-3$ is included in the solution set.

Shade the number line to the left of $-3$ because $x$ must be less than or equal to $-3$, which means all values less than $-3$ are part of the solution.

Label the graph clearly, showing the shaded region and the solid circle at $-3$, to represent the solution to the inequality $x \leq -3$.

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Key Concepts

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

Graphing Inequalities on a Number Line

Graphing inequalities involves representing all values that satisfy the inequality on a number line. For example, x ≤ -3 includes all numbers less than or equal to -3, shown by shading to the left of -3 and using a solid dot at -3 to indicate inclusion.

Understanding Inequality Symbols

Inequality symbols like ≤ mean 'less than or equal to.' This indicates that the solution set includes the boundary value and all values less than it. Recognizing these symbols helps determine which side of the number line to shade.

Closed and Open Dots on Graphs

A closed dot on a number line indicates that the endpoint is included in the solution (e.g., ≤ or ≥), while an open dot means the endpoint is excluded (e.g., < or >). This distinction is crucial for accurately representing inequalities.

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Textbook Question

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.

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