Problem 19 - Algebra 2 and Trigonometry

Question

In \(15-26,\) find and graph the solution set of each inequality. $$ |-5-a| > 4 $$

Step-by-Step Solution

Answer

The solution set is \(a < -9\) or \(a > -1\).

Step 1: Understand Absolute Value Inequality

The inequality \(|x| > c\) implies two separate inequalities, either \(x > c\) or \(x < -c\). The expression inside the absolute value can be greater or less than the positive threshold.

Step 2: Apply to Inequality

Given \(|-5 - a| > 4\), apply the logic for absolute values: \(-5 - a > 4\) or \(-5 - a < -4\). These represent the conditions for when the absolute value part will exceed 4, either positively or negatively.

Step 3: Solve First Inequality

For \(-5 - a > 4\): 1. Add 5 to both sides: \(-a > 9\),2. Multiply by -1 to solve for \(a\) (and flip inequality): \(a < -9\).

Step 4: Solve Second Inequality

For \(-5 - a < -4\): 1. Add 5 to both sides: \(-a < 1\),2. Multiply by -1 to solve for \(a\) (and flip inequality): \(a > -1\).

Step 5: Write Solution Set

Combine the results from Steps 3 and 4 to find that the solution set is \(a < -9\) or \(a > -1\). This means any \(a\) that meets either condition is part of the solution set.

Step 6: Graph the Solution Set

Graph the solution on a number line by:1. Marking an open circle on -9 and extending a line to the left, representing \(a < -9\).2. Marking an open circle on -1 and extending a line to the right, representing \(a > -1\). The graph illustrates that \(a\) can be any value less than -9 or greater than -1.

Key Concepts

Solution Set

A solution set refers to the collection of all possible values that satisfy the given inequality. In absolute value inequalities, like \(|-5-a| > 4\), the solution set can often be split into two separate inequalities. This is due to the nature of absolute value, which measures distance from zero. Therefore, it inherently produces two conditions - one for each direction.

For the inequality provided, \

• Start by considering \(-5-a > 4\), which simplifies to \(a < -9\).
• Simultaneously, consider \(-5-a < -4\), which simplifies to \(a > -1\).

Combining these inequalities yields the solution set \(a < -9\) or \(a > -1\). This means that any number less than -9 or greater than -1 satisfies the inequality \(|-5-a| > 4\).

The word "or" here is crucial, signifying that if either condition is true, then the overall inequality holds true. Always ensure to separate your solutions when dealing with "greater than" scenarios in absolute inequalities.

Graphing Inequalities

Graphing inequalities provides a visual way to understand the solution set of an inequality. It helps us see the range of possible solutions at a glance. For our solution set, \(a < -9\) or \(a > -1\), we represent these on a number line.

• Start with \(a < -9\). Place an open circle at -9. Draw a line extending to the left to show that every number less than -9 is a solution.
• Next, graph \(a > -1\). Place another open circle at -1. This time, draw a line extending to the right, indicating that every number greater than -1 is a solution.

Open circles are used here to show that the endpoints, -9 and -1, are not included in the solution set. This is typical when dealing with strict inequalities, represented by symbols such as \(>\) or \(<\).
The graph helps reinforce the understanding that these separate intervals on the number line cover all the values that satisfy the original inequality.

Number Line Representation

A number line is a powerful tool for showing solutions to inequalities like \(|-5-a| > 4\). Using a number line representation, we can clearly visualize where the solution set lies.

• First, number lines allow us to label points of interest, like -9 and -1 in our case.
• We use open circles for these points because the original inequalities, \(a < -9\) and \(a > -1\), do not include -9 and -1 themselves.
• The lines extending from these open circles signify that all numbers in those directions are solutions.

This visual method complements the algebraic steps we took to find the solution set. Seeing the intervals physically helps confirm that values less than -9 and greater than -1 are indeed solutions.
Remember, visualizing with a number line can often make complex algebraic solutions more intuitive and easier to grasp.