Problem 58 - 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. $$ |x+2| \geq 5 $$

Step-by-Step Solution

Answer

The solution is \((\infty, -7] \cup [3, \infty)\).

Step 1: Enter Equation into Graphing Utility

Input \(Y_1 = |x+2|\) and \(Y_2 = 5\) into your graphing calculator. This sets up the inequality equation for graphing.

Step 2: Access Absolute Value Function

To enter the absolute value expression for \(Y_1\), go to the MATH menu, select Num, and choose 1:abs(. This is where you input \(x+2\).

Step 3: Graph the Equations

Once \(Y_1\) and \(Y_2\) are entered, graph both equations on the same axes. This visual representation helps us find the intersection points.

Step 4: Find Intersection Points

Access the calculation menu by pressing \(2^{nd}\) CALC. Choose option 5:intersection. For '1st curve' and '2nd curve', press enter without changing anything. After 'guess', press enter again. Record the \(x\)-values of the intersection points.

Step 5: Analyze the Inequality

For \(|x+2| \geq 5\), we solve both \(x+2 \geq 5\) and \(-(x+2) \geq 5\). The solutions are \(x \geq 3\) and \(x \leq -7\). These intervals correspond to where the graph of \(Y_1\) is above or equal to \(Y_2\).

Step 6: Sketch and Shade Solution Set

Sketch the graph showing two lines intersecting at the points found. Shade the regions on the \(x\)-axis where \(x \geq 3\) and \(x \leq -7\), which represent the solution set where \(|x+2| \geq 5\).

Step 7: Write Answer in Interval Notation

The solution set where the inequality holds true is \(x \leq -7\) and \(x \geq 3\). Therefore, in interval notation, the answer is \((\infty, -7] \cup [3, \infty)\).

Key Concepts

Using a Graphing Calculator for Absolute Value Inequalities

Graphing calculators are extremely useful when dealing with absolute value inequalities, such as \(|x+2| \geq 5\). To solve this using a graphing calculator, we first need to input the expressions separately as different graphs. Here’s how you can easily do this:

• First, place the left-hand side of the inequality as a function, say \(Y_1 = |x+2|\).
• Next, input the constant from the right-hand side, \(Y_2 = 5\).
• Access the MATH menu by navigating to 'Num' and selecting option 1:abs(. This is where you input \(x+2\).

Once these are set up, graph both functions on the same axes. This will produce a visual representation that shows the behavior of both expressions and allows you to find crucial intersection points. At intersection points, both expressions hold the same value, which is essential for solving the inequality.

Using Interval Notation in Solution Set Notation

Interval notation is a concise way of expressing a range of numbers. When solving inequalities such as \(|x+2| \geq 5\), interval notation can help clearly communicate the solution set. After finding where the inequality holds true, either graphically or analytically, the solutions might look like:

• \(x \geq 3\)
• \(x \leq -7\)

In mathematical language, these ranges are represented using interval notation:

• For \(x \geq 3\), the interval is \([3, \infty)\), indicating that \(x\) can be any number from 3 onwards.
• For \(x \leq -7\), it is \((\infty, -7]\), indicating all values less than or equal to -7.

Interval notation provides a precise way to express solutions efficiently, incorporating any number within the fixed intervals without explicitly listing out numbers.

Finding Intersections in Graphs

Identifying points where two graphs intersect is crucial when dealing with inequalities, since these intersections often reveal solutions to equations. When two graphs share a common point, this implies the two expressions are equal at these points. Here’s how to find these intersections using a graphing utility:

• Access the calculation tools by pressing \(2^{nd}\) CALC on your graphing calculator.
• Choose option 5:intersection from the menu.
• Confirm the first and second curves by pressing enter when prompted. The calculator wants to ensure which graphs you're referring to.
• It will prompt you for a guess; press enter again to let the calculator find it.
• The \(x\)-values at the intersections will signify boundary points for the inequality's solution.

Understanding where and why these intersections happen simplifies solving inequalities, because they mark transitions where one graph’s value starts being above or below the other. Consider them as thresholds on a number line, critical for knowing where the inequality reverses.