Problem 55 - College Algebra

Question

Input the left-hand side of the inequality as a \(\mathrm{Y} 1\) 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, 1:abs(. Find the points of intersection, recall ( \(2^{\text {nd }}\) CALC 5:intersection, 1st 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: \((-\infty, \infty)\).

Step 1: Rewrite the Inequality

The given inequality is \(-\frac{1}{2}|x+2|<4\). To make it easier to graph, we first eliminate the negative factor by dividing both sides of the inequality by -1/2, changing the direction of the inequality: \(|x+2| > -8\). Since the expression \(-8\) is negative and absolute values are always non-negative, this inequality is always true for all real numbers \(x\).

Step 2: Identify the Y1 and Y2 Functions

For graphing purposes, input \(Y1 = -\frac{1}{2}|x+2|\) and \(Y2 = 4\) into the graphing calculator. This will allow us to visualize the inequality as intersections of the graphs \(Y1\) and \(Y2\).

Step 3: Input the Equations into the Graphing Calculator

Press the 'Y=' button on the calculator. Enter \('-\frac{1}{2}'\) followed by 'MATH', 'NUM', '1:abs(', and input 'X+2)' for \(Y1\). Enter \(4\) for \(Y2\).

Step 4: Determine the Intersection Points

Use the intersection function by pressing '2nd', 'CALC', and choosing '5:intersect'. Select the first curve (Y1) with 'ENTER', the second curve (Y2) with 'ENTER', and a guess using the directional arrows, followed by 'ENTER'. The calculator will show the intersection point(s).

Step 5: Sketch the Graph and Shade the Solution

Draw a sketch of the graph showing \(Y1 = -\frac{1}{2}|x+2|\) and \(Y2 = 4\). Shade the portion of the \(x\)-axis where the graph of \(Y1\) is below the graph of \(Y2\).

Step 6: Write the Solution in Interval Notation

The solution to the inequality \(-\frac{1}{2}|x+2| < 4\) is all real numbers because the absolute value inequality was made greater than a negative number, which is always true. In interval notation, the solution is \((-\infty, \infty)\).

Key Concepts

Absolute Value

Absolute value is a fundamental concept in mathematics that represents the distance of a number from zero on the number line, without considering which direction the number lies. This means that the absolute value of a number is always non-negative. For any real number \(x\), the absolute value is denoted by \(|x|\). Here are some key points to understand about absolute value:

• If \(x\) is positive or zero, \(|x| = x\).
• If \(x\) is negative, \(|x| = -x\), which is the positive counterpart of \(x\).

In the context of inequalities, the absolute value has a significant role. For example, the expression \(|x+2|\) signifies how far \(x + 2\) is from zero. This is crucial when solving inequalities like the one given in the exercise, where transforming the inequality allowed us to better visualize its solution.

Graphing Utilities

Graphing utilities, such as graphing calculators or software tools, are invaluable for visualizing mathematical functions and their interactions. These tools provide visual representations that help in understanding the relationships between different mathematical expressions. Here’s how graphing utilities were used in the exercise:

• The inequality \(-\frac{1}{2}|x+2| < 4\) was input into a graphing calculator to visualize \(Y1\) and \(Y2\) as functions.
• By graphing \(Y1 = -\frac{1}{2}|x+2|\) and \(Y2 = 4\), the points of intersection were identified, which are crucial for understanding solutions to inequalities.

Graphing utilities simplify the process of finding intersection points, which can otherwise be complex algebraically. This visual aid can help students affirm their algebraic solutions by providing a pictorial preview of the problems they are solving.

Interval Notation

Interval notation is a method of writing subsets of the real number line. It is a concise way of specifying a range of values that satisfy a given inequality. Here are the basics of interval notation:

• Brackets \([\ ]\) are used when the endpoints are included in the interval (closed interval).
• Parentheses \((\ )\) are used when the endpoints are not included (open interval).
• If an interval extends to infinity, it is always open because infinity is not a number that can be reached or included.

In the solution of the exercise, the interval \(( -\infty, \infty )\) represents all real numbers. This outcome arose because the inequality involving an absolute value produces a condition that is true for any possible real number. Understanding how to express solution sets in interval notation helps in clearly communicating the scope of the values that satisfy a given inequality.