Problem 5 - Algebra 2

Question

Solve each inequality. Graph the solution set on a number line. $$ |a| \geq 5 $$

Step-by-Step Solution

Answer

The solution is \(a \leq -5\) or \(a \geq 5\).

Step 1: Understanding absolute value inequality

To solve the inequality \(|a| \geq 5\), it's important to understand that this implies two separate inequalities: \(a \geq 5\) or \(a \leq -5\). This is because the absolute value \(|a|\) refers to the distance of \(a\) from 0 on the number line, which is at least 5 units away.

Step 2: Solve the first inequality

Consider the inequality \(a \geq 5\). Since \(a\) needs to be greater than or equal to 5, the values of \(a\) would be all numbers starting from 5 and extending to positive infinity.

Step 3: Solve the second inequality

Now, consider the inequality \(a \leq -5\). This means that \(a\) is any number less than or equal to -5, thus extending from negative infinity to -5.

Step 4: Combine the solutions

The solution to the inequality \(|a| \geq 5\) involves combining the solutions from Steps 2 and 3. Hence, the solution set is \(a \leq -5\) or \(a \geq 5\).

Step 5: Graph the solution on a number line

To graph the solution, draw a number line. Place a closed circle on -5 and extend the line to the left to represent \(a \leq -5\). Similarly, place a closed circle on 5 and extend the line to the right for \(a \geq 5\). This visually represents the solution set on the number line.

Key Concepts

Number Line Graphing

Graphing inequalities on a number line is a visual way to represent solution sets. For the inequality \(|a| \geq 5\), we represent two conditions: \(a \leq -5\) and \(a \geq 5\).
To graph these:

• For \(a \leq -5\), draw a number line and place a closed circle at -5. Then, shade all the numbers to the left, representing all values less than or equal to -5.
• For \(a \geq 5\), place another closed circle at 5. Shade to the right of this point, capturing all numbers greater than or equal to 5.

This method highlights how the solution set covers two separate intervals on the number line, visually showing the absolute value concept of distance from zero.

Inequality Solutions

Solving inequalities involves finding all possible values of a variable that satisfy the given conditions. With \(|a| \geq 5\), we split this into two inequalities: \(a \geq 5\) and \(a \leq -5\). These arise from understanding that the absolute value of \(a\) must be at least 5 units from zero.

• Solving \(a \geq 5\) simply means finding values of \(a\) starting from 5 onward, covering all larger numbers.
• For \(a \leq -5\), solutions are numbers starting from -5 and stretching leftwards towards negative infinity.

This breakdown reflects how an absolute value inequality can define a range of solutions instead of a single point.

Absolute Value Concepts

Absolute value represents the distance of a number from zero, regardless of direction. For \(|a| \geq 5\), this means the value of \(a\) has to be at least 5 units away from zero, whether on the positive or negative side.
Key concepts include:

• Absolute values are always non-negative since they measure distance.
• \(|a|\) translates into two scenarios: \(a ≥ 5\) or \(a ≤ -5\), covering either side of zero.
• This distance interpretation helps in understanding why inequalities might result in two-part solutions on a number line.

Visualizing these principles through a number line can simplify understanding how absolute values define broad solution sets.