Problem 54 - 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. $$ |x+2|-5 < 2 $$

Step-by-Step Solution

Answer

The solution is (-9, 1) in interval notation.

Step 1: Input Y1 and Y2 in Graphing Utility

Start by entering the left-hand side of the inequality into Y1 as \(|x+2|-5\), and the right-hand side into Y2 as \(2\). Use the absolute value function from the MATH menu.

Step 2: Find Points of Intersection

Use the graphing utility's 2nd CALC function and select the 5th option, 'intersection'. Follow the instructions for the first curve, second curve, and make a guess to find the points where the two graphs intersect.

Step 3: Determine the Intersection Points

Based on the calculation in the graphing utility, note the x-values where the two graphs intersect. These represent the points of equality \(|x+2|-5 = 2\).

Step 4: Analyze the Inequality

Assess the intervals created by the intersection points to determine where the inequality \(|x+2|-5 < 2\) holds true. You will consider the segments of the x-axis where the Y1 is less than Y2.

Step 5: Write the Solution in Interval Notation

Shade the interval on the x-axis that satisfies the inequality and express this solution using interval notation.

Key Concepts

Absolute Value Functions

Absolute value functions are fascinating in mathematics as they help us measure distance from zero without considering the direction. When you see an expression like \(|x+2|\), it calculates how far \(x+2\) is from zero. Essentially, it always gives a non-negative result, regardless of whether \(x+2\) is positive or negative.

In our exercise, we need to handle \(|x+2|-5 < 2\). This means we want to find x-values where the expression inside stays less than 2 when we've subtracted 5 from the absolute value. Absolute value functions often create two scenarios:

• If the contents inside the absolute value are positive or zero, the function behaves like any linear function.
• If negative, the absolute flips it to its positive equivalent, which can create unique graphical behavior.

Understanding these properties allows us to break the inequality into different segments and solve each based on their interval.

Interval Notation

Interval notation is a convenient way to write the solution sets of inequalities, especially useful in describing intervals on the x-axis. It succinctly tells you where a function resides in relation to a specified inequality.

Let's consider our inequality \(|x+2|-5 < 2\). Once you find the x-values where it holds true, you can express them in interval notation, which uses brackets and parentheses for inclusivity and exclusivity:

• Parentheses \(( )\) indicate that the value is not included in the interval.
• Brackets \([ ]\) mean the value is included.

For example, if the solution is between -3 and 1, the interval in notation would be \((-3, 1)\), meaning \(x\) can be any number between -3 and 1 but not exactly -3 or 1. The ability to express solutions clearly in terms of intervals is crucial for understanding the range of values that satisfy an inequality.

Graphing Utility

Using a graphing utility is an excellent way to visualize complex inequalities like \(|x+2|-5 < 2\). These powerful tools allow you to graph equations and inequalities, making understanding them far easier.

Here's how to effectively use a graphing utility:

• Input the expressions into the Y-values. For our inequality, enter \(|x+2|-5\) into \(Y1\) and \(2\) into \(Y2\).
• Use the graph to observe where these two lines meet. The intersection points are crucial as they define where the left side equals the right side, or \(|x+2|-5 = 2\).
• Use the calculator or software's intersection finding tools, like the "2nd CALC" and "intersection" function, to pinpoint these exact x values.

By examining the graph visually, you can easily determine where the inequality holds by identifying the interval where Y1 lies below Y2. This approach transforms abstract algebraic problems into more tangible visuals, helping you better understand and solve inequalities.