Problem 16 - Algebra 2 and Trigonometry

Question

In \(15-26,\) find and graph the solution set of each inequality. $$ |a-5| \geq 3 $$

Step-by-Step Solution

Answer

The solution set is \(a \leq 2\) or \(a \geq 8\).

Step 1: Understanding Absolute Inequality

The inequality \(|a - 5| \geq 3\) suggests that the expression inside the absolute value, \(a-5\), can have values either less than or equal to -3 or greater than or equal to 3. This inequality prompts us to break it into two separate inequalities: \(a - 5 \leq -3\) or \(a - 5 \geq 3\).

Step 2: Solving the First Inequality

Let's solve the inequality \(a - 5 \leq -3\). Adding 5 to both sides gives: \[ a \leq 2 \]

Step 3: Solving the Second Inequality

Now, solve the second part of the inequality \(a - 5 \geq 3\). Adding 5 to both sides yields: \[ a \geq 8 \]

Step 4: Combining the Solutions

The solutions to \(|a - 5| \geq 3\) are the values of \(a\) that satisfy either \(a \leq 2\) or \(a \geq 8\). The solution set is therefore: \[ a \leq 2 \text{ or } a \geq 8 \]

Step 5: Graphing the Solution on a Number Line

On a number line, graph two separate ranges for the solution set. Plot a closed circle at \(a = 2\) and shade everything to the left to show \(a \leq 2\). Then, plot another closed circle at \(a = 8\) and shade everything to the right to illustrate \(a \geq 8\). This graphically represents the solution set \(a \leq 2 \cup a \geq 8\).

Key Concepts

Inequality Solutions

Inequality solutions involve finding the set of values that satisfy the given inequality. This process can be directly tied to how equations are solved, but with some extra rules for inequalities. In the case of absolute inequality, such as \(|a - 5| \geq 3\), we separate it into two parts due to the properties of absolute values.

• The expression inside the absolute value can be equal to the given value or more, therefore covering two possible conditions:
  • For the example \(|a - 5| \geq 3\):
    • One condition is \(a - 5 \leq -3\)
    • The other is \(a - 5 \geq 3\)
• Each of these parts is solved individually like regular inequalities.

This leads us to two segments of solutions, that tell us in which ranges the inequality is true. Recognizing and separating the problem into components helps in forming a comprehensive solution.

Graphing Inequalities

Graphing inequalities is a visual way to present the solution of an inequality on a number line. This approach lets you easily see the available range of values without needing to go through solutions numerically.
For the inequality \(|a - 5| \geq 3\), we found two solution ranges: \(a \leq 2\) and \(a \geq 8\). Here's how to represent these graphically:

• Place a closed circle on the number line at number 2 and shade everything to the left - this darkened section shows all numbers less than or equal to 2.
• Similarly, place a closed circle at number 8 and shade everything to the right - demonstrating numbers greater than or equal to 8.

The circles indicate that the endpoints (2 and 8) are included in the solution set, aligning with the inequalities "less than or equal" and "greater than or equal." This visualization aids students in understanding the scope of solutions and how inequalities unfold across a number line.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators. They are the building blocks for equations and inequalities that frequently appear in algebra. In our exercise \(|a - 5| \geq 3\), the expression \(a - 5\) is central.

• Here, \(a\) is a variable, and 5 is a constant.
• The absolute value operator \(| \cdot |\) emphasizes distance from zero, making calculations always positive or zero.

By manipulating algebraic expressions such as \(a - 5\), students learn to isolate variables, understand operations, and solve problems involving multiple parts.
The knowledge of interpreting and working with these expressions is foundational to developing skill in solving equations and inequalities. This allows more complex algebraic ideas to be handled with confidence and accuracy in future problems.