Problem 55 - College Algebra

Question

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter \(\mathrm{Y} 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, liabs ( Find the points of intersection, recall \((2^{\text { nd }}\) CALC 5:intersection, lst curve, enter, } \(2^{\text { nd }}\) curve, enter, guess, enter). Copy a sketch of the graph and shade the \(x\) -axis for your solution set to the inequality. Write final answers in interval notation. $$ -\frac{1}{2}|x+2|<4 $$

Step-by-Step Solution

Answer

The solution is all real numbers, written as \((- \infty, \infty)\).

Step 1: Rewrite the inequality

The given inequality is \(-\frac{1}{2}|x+2|<4\). First, isolate the absolute value expression by rewriting the inequality as \[ |x+2| > -8 \]. Since absolute values are always non-negative, the inequality \(|x+2| > -8\) is always true for all real numbers.

Step 2: Graph the equation

To begin solving graphically, input the left side \(Y_1 = -\frac{1}{2}|x+2|\) and the right side \(Y_2 = 4\) into a graphing calculator. The absolute value function can be entered via the MATH menu under Num to \lst in the calculator.

Step 3: Find the points of intersection

Use the graphing calculator's intersection feature (2nd CALC, then option 5) to find where \(Y_1\) and \(Y_2\) intersect. Follow the prompt to select the first curve, second curve, and provide a guess. However, in this specific case, intersections do not need to be calculated because the inequality is always satisfied.

Step 4: Solution set on graph

According to the graphical representation and algebraic interpretation, the line \(Y_1\) never intersects with \(Y_2\) as a solution since the inequality \(|x+2| > -8\) holds for all \(x\). Hence, the entire x-axis is part of the solution.

Step 5: Write solution in interval notation

Since the inequality holds for all real numbers, the solution in interval notation is \((-\infty, \infty)\) indicating all real numbers are solutions.

Key Concepts

Absolute Value

Absolute value is a crucial mathematical concept that measures the distance of a number from zero on the number line. It's represented by two vertical bars, such as \(|x|\). The absolute value of any number is always non-negative, which means it can only be zero or positive.
When dealing with expressions like \(|x + 2|\), you're considering the distance between \(x + 2\) and zero. Whether \(x + 2\) is positive or negative, the absolute value will be the positive equivalent.

• Example: \(|-3| = 3\)
• Usage: Often used in inequalities and distance calculations.

In our exercise, isolating the absolute value expression helps in interpreting the inequality, \(|x + 2| > -8\), showing that every real number satisfies the condition because absolute values can't be negative.

Interval Notation

Interval notation is a shorthand used in mathematics to describe a range of numbers. It's a convenient way to express a collection of solutions without listing every number. We commonly use parentheses "()" and brackets "[]" to define these intervals.

• Parentheses \((\) are used for numbers that are not included in the interval (open interval).
• Brackets \([\) indicate that the number is included in the interval (closed interval).

In our example, the solution for the inequality \(|x + 2| > -8\) is written in interval notation as \((-\infty, \infty)\). This denotes that all real numbers are solutions.
This notation efficiently communicates that any number you choose is valid without needing to state each individually.

Graphing Inequalities

Graphing inequalities involves plotting functions on a coordinate plane to visually represent the range or scope of solutions. By using graphs, we can easily understand where one function is greater or less than another.
If you have an absolute value inequality, like our given exercise, graphing each side of the inequality separately helps to visualize their relationships:

• \(Y_1 = -\frac{1}{2}|x+2|\)
• \(Y_2 = 4\)

By using a graphing calculator, you can plot these functions and observe their behavior. In our case, the graph shows that \(Y_1\) is always below \(Y_2\), confirming that the inequality holds for all values of \(x\). This method simplifies checking complex inequalities by giving a clear visual representation.

Real Numbers

Real numbers include all the numbers you can think of: positive, negative, whole numbers, fractions, and decimals. Essentially, all the numbers that you can locate on a number line are real numbers.

• Positive numbers: 1, 2.5, 100.
• Negative numbers: -1, -2.5, -100.
• Zero.
• Fractions and decimals, like \(\frac{1}{2}\) or 0.75.

In the context of our exercise, understanding that all real numbers satisfy the condition \(|x + 2| > -8\) is important. Because absolute values are never negative, this inequality is inherently true for the entire set of real numbers, thus covering every point on the number line.
Recognizing that \((-\infty, \infty)\) corresponds to the set of all real numbers helps to confirm the solution set of an inequality problem.