Problem 58 - College Algebra with Corequisite Support

Question

For the following exercises, input the left-hand side of the inequality as a \(Y 1\) graph in your graphing utility. Enter \(y 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, \(1: a b s(.\) Find the points of intersection, recall \(\left(2^{\text {nd }}\right.\) CALC 5 :intersection, \(1^{\text {st }}\) 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 (-∞, -7] ∪ [3, ∞) in interval notation.

Step 1: Understand the Inequality

The inequality given is \(|x+2| \geq 5\). This means we need to find the set of x-values for which the expression inside the absolute value, plus 2, is either greater than or equal to 5, or less than or equal to -5.

Step 2: Set Up Graph in Graphing Utility

To solve this inequality graphically, we input the left-hand side of the inequality as \(Y_1 = |x+2|\) and the right-hand side as \(Y_2 = 5\) into a graphing calculator or utility.

Step 3: Graph the Functions

After inputting the equations, graph both functions. The graph of \(Y_1 = |x+2|\) is a V-shaped graph, moved two units to the left. The graph of \(Y_2 = 5\) is a horizontal line at \(y = 5\).

Step 4: Find Points of Intersection

Using the graphing calculator's intersection function (\(2^{\text{nd}} \) CALC 5 :intersection), find where \(Y_1\) and \(Y_2\) intersect. The possible intersection points are where \(x+2 = 5\) and \(x+2 = -5\).

Step 5: Solve for Intersection Points

Calculate the intersection points: - For \(x + 2 = 5\), solve for \(x\) to get \(x = 3\). - For \(x + 2 = -5\), solve for \(x\) to get \(x = -7\).

Step 6: Determine the Solution Set

The inequality \(|x+2|\geq 5\) holds true for values of \(x\) where the graph of \(Y_1\) is on or above the graph of \(Y_2\). Thus, it holds true in the intervals \(x \leq -7\) and \(x \geq 3\). These intervals correspond to where the V-shaped graph is either at or above the horizontal line.

Step 7: Write Answer in Interval Notation

The solution to the inequality, expressed in interval notation, is (-\infty, -7] \cup [3, \infty). This shows the range of \(x\)-values for which the inequality is satisfied.

Key Concepts

Absolute Value

Absolute value is an essential concept in mathematics that represents the distance of a number from zero on the number line. It is always non-negative. For instance, the absolute value of -3 is 3, simply written as \(|-3| = 3\). The expression around the absolute value bars \( |x+2| \) denotes the distance of \(x+2\) from zero. This means the expression \( |x+2| \geq 5 \) asks for the values of \(x\) where the distance from -2 is at least 5. This leads to two conditions:\
\

\
• \(x + 2 \geq 5\)
\
• \(x + 2 \leq -5\)
\

\When solving absolute value inequalities, break them down into these two separate inequalities. Solve each one and consider their solutions together, as absolute values denote a range of values, beyond just positive solutions.

Graphing Utility

A graphing utility is a powerful tool used to visualize mathematical functions and solve equations or inequalities graphically. Tools like graphing calculators or software allow you to plot graphs and find points of intersection easily. \
\To solve the inequality \(|x+2| \geq 5\) using a graphing utility, follow these steps:\
\

\
• Input the expression \(Y_1 = |x+2|\) as the first function. This will display a V-shaped graph on the graphing tool.
\
• Input the constant \(Y_2 = 5\) as a horizontal line.
\
• Use the intersection feature, typically found under a 'CALC' menu, to determine where these two lines meet. This helps identify the critical values where the inequality switches from false to true or vice versa.
\

\Graphing is unparalleled for understanding inequalities visually. It quickly allows you to see where one function is greater than or equal to another, helping to grasp solution intervals intuitively.

Interval Notation

Interval notation is a concise way to express a set of real numbers. It is especially useful for representing solutions of inequalities. The notation includes brackets and parentheses to show which endpoints are included or excluded in the interval. \
\For example, consider the solution to \(|x+2| \geq 5\). Solving this results in two intervals: \(x \leq -7\) and \(x \geq 3\).\
\

\
• \((-\infty, -7]\) means that \(x\) can be any number less than or equal to -7, extending indefinitely to the left.
\
• \([3, \infty)\) indicates that \(x\) is any number greater than or equal to 3, extending indefinitely to the right.
\
• The union symbol \(\cup\) combines these intervals to represent all possible solution values of \(x\) for which the inequality holds true.
\

\Interval notation is neat and clear, providing an efficient way to communicate solutions without needing verbose descriptions or separate sets.