Problem 24 - Algebra 2 and Trigonometry

Question

In \(15-26,\) find and graph the solution set of each inequality. $$ \left|\frac{5}{2} x+2\right| \leq 0 $$

Step-by-Step Solution

Answer

The solution set is \( x = -\frac{4}{5} \), graphed as a single point at \( x = -\frac{4}{5} \).

Step 1: Understand the Structure of Absolute Value Inequality

The absolute value inequality given is \( \left|\frac{5}{2} x+2\right| \leq 0\). The key understanding with absolute values is that they are non-negative. Therefore, \( |A| \leq 0 \) means \( A = 0 \). This is because absolute values can't be negative.

Step 2: Set the Expression Inside the Absolute Value to Zero

Since \( \left|\frac{5}{2} x + 2\right| \leq 0 \) implies \( \frac{5}{2} x + 2 = 0 \), we solve for \( x \) by setting the expression inside the absolute value to zero.

Step 3: Solve the Equation

Solve the equation \( \frac{5}{2} x + 2 = 0 \).Subtract 2 from both sides:\[ \frac{5}{2} x = -2 \]Multiply both sides by \( \frac{2}{5} \) to solve for \( x \):\[ x = -\frac{2 \cdot 2}{5} x = -\frac{4}{5} \]

Step 4: Check the Solution

Plug \( x = -\frac{4}{5} \) back into the expression to verify:\( \frac{5}{2}(-\frac{4}{5}) + 2 = -2 + 2 = 0 \)The left side equals zero, confirming the solution is correct.

Step 5: Graph the Solution Set

Since the inequality holds true only for \( x = -\frac{4}{5} \), the graph is a single point at \( x = -\frac{4}{5} \) on the number line. This means the solution set is \( x = -\frac{4}{5} \), represented by a closed dot at this point.

Key Concepts

Inequality Solutions

Inequality solutions involve finding all possible values of a variable that make an inequality true. Unlike equations, which specify exact values, inequalities can cover a range or set of values. However, in some special cases, like the inequality \( |A| \leq 0 \), they can pinpoint a specific value. For an inequality involving absolute values, the solution often requires setting the expression inside the absolute value equal to zero.

Here is why: The absolute value expression \(|A|\) is always zero or positive. So, when asked to solve \(|A| \leq 0\), the only possibility is that \(A\) itself must be zero. Solving for \(x\) then becomes the focus of the exercise.

• If the expression within the absolute value results in zero, then the inequality holds only for that specific value of the variable.
• For more common inequalities, such as \(|A| < b\) or \(|A| > b\), the solution set provides a range of possible values.

Understanding these steps helps identify whether the solution is a range or a specific value.

Absolute Value Equations

Absolute value equations are concerned with expressions within absolute value bars. An absolute value denotes the distance of a number from zero on the number line, always resulting in a non-negative value.

When dealing with equations like \( \left|\frac{5}{2} x + 2\right| = 0 \), it's important to realize that the absolute value being zero occurs only when its contents are zero. Thus, the expression \(\frac{5}{2} x + 2\) equals zero.

• To solve, set the equation inside the absolute value equal to zero and solve for \(x\).
• Subtract, add, or multiply to isolate \(x\).

By understanding this principle, students can more easily tackle absolute value equations, applying the same strategy across a range of problems.

Graphing Inequalities

Graphing inequalities involves visually representing the solution set on a number line. For the equation \( \left|\frac{5}{2} x + 2\right| \leq 0 \), we determined that \(x = -\frac{4}{5}\) is the only solution.

In this case, the graph on the number line is just a single point. Here's how graphing would proceed:

• First, identify the solution from solving the inequality or equation.
• Place a closed dot on the number line at \(-\frac{4}{5}\) to denote the solution's inclusion.
• A closed dot signifies that the endpoint itself is part of the solution set.
• No shading is required here since the solution is not a range.

Visual representations assist in confirming and understanding solution sets' positional context, simplifying interpretation of outcomes.