Problem 12 - Precalculus: Mathematics for Calculus

Question

Graph the inequality. $$x \leq-1$$

Step-by-Step Solution

Answer

Shade to the left of -1 and use a filled dot on -1.

Step 1: Identify the inequality

The inequality given is \( x \leq -1 \). This means that \( x \) is any number less than or equal to \(-1\).

Step 2: Draw the number line

Draw a horizontal line, which represents the number line. It should have numbers marked on it at equal intervals. Ensure that the point \(-1\) is clearly marked on this line.

Step 3: Plot the critical point

Locate the point \(-1\) on the number line. Since the inequality is \( \leq \) (less than or equal to), we will use a closed dot or filled circle at point \(-1\) to show that \(-1\) is included in the solution.

Step 4: Shade the appropriate region

Shade the area to the left of the point \(-1\) on the number line. This shaded area shows all values that are less than \(-1\), which are included in the solution of the inequality \( x \leq -1 \).

Key Concepts

Graphing Inequalities

Graphing inequalities helps you visualize the set of solutions that fulfill the inequality. When dealing with inequalities in one variable, like \( x \leq -1 \), the solution is graphically represented on a number line. The graph includes all possible values of \( x \) that make the inequality true.

• Identifying the Type: In the inequality \( x \leq -1 \), the symbol \( \leq \) means "less than or equal to." This tells us to include the number -1 and all numbers less than -1 in our solution set.
• Representation: Use a closed dot on the graph to show that -1 is part of the solution. This indicates clear inclusion on the number line graph.

By following these steps, you can turn a math statement into a visual graph that highlights all solutions.

Number Line

A number line is a straight, horizontal line used to represent numbers at equal intervals. This tool is essential for graphing inequalities because it allows a clear visualization of numerical relationships.

• Structure: The number line includes zero and extends infinitely in both the positive and negative directions.
• Intervals: Equal spacing between numbers helps accurately place critical points and shade regions for inequalities.

On the line, the point -1 would be positioned based on the selected scale, marking this point helps identify the specific area to shade. Always ensure your number line is clear and includes necessary numerical markings to correctly graph an inequality.

Inequality Solutions

Solutions to inequalities like \( x \leq -1 \) include all values of \( x \) that make the inequality true.

• Inclusion Criteria: The expression "less than or equal to" means -1 is included in the solutions, as are all numbers to its left.
• Notation: Use brackets or filled dots to indicate inclusion. For instance, with \( \leq \), use a filled circle on that point.

The shaded region on the number line represents all numbers fulfilling the inequality. Ensure that the shading is accurate, as it visually demonstrates solutions for anyone reviewing the graph.

Critical Points

Critical points are specific values on the number line that determine boundary conditions of the inequality solution. In the inequality \( x \leq -1 \), the critical point is -1.

• Significance: This point is crucial as it marks the transition between included and excluded values.
• Graphical Indication: A closed dot is placed at -1 to signify that it is part of the solution set. This differentiates it clearly from an open dot used for \( < \) or \( > \).

Identify the critical points first when graphing as they guide where to place marks and shading. A correct determination of critical points prevents errors in understanding what values satisfy the inequality.