Ch. 5 - Systems and Matrices

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 5 - Systems and Matrices

Problem 19

Chapter 6, Problem 19

Graph each inequality. x ≤ 3

Verified step by step guidance

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 corresponding to $x = 3$ on it.

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

Shade the number line to the left of $x = 3$ because all values less than 3 satisfy the inequality.

Label the shaded region clearly to show that it represents all $x$ such that $x \leq 3$.

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Key Concepts

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

Inequalities on the Number Line

An inequality like x ≤ 3 represents all values of x that are less than or equal to 3. On a number line, this includes the point 3 and all points to its left. Understanding how to represent these values visually is essential for graphing inequalities.

Closed and Open Circles

When graphing inequalities, a closed circle is used to indicate that the endpoint is included (≤ or ≥), while an open circle shows the endpoint is excluded (< or >). For x ≤ 3, a closed circle is placed at 3 to show that 3 is part of the solution.

Shading the Solution Region

After marking the boundary point, the solution region is shaded to represent all values satisfying the inequality. For x ≤ 3, the shading extends to the left of 3, indicating all numbers less than or equal to 3 are solutions.

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