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

Step-by-Step Solution

Answer

The solution is \((-9, 5)\).

Step 1: Transform Inequality into Equation

First, we convert the inequality \(|x+2|-5<2\) into an equation to find the points of intersection. This becomes \(|x+2|-5=2\). We'll solve this equation to help identify points on the graph.

Step 2: Solve the Equation for Intersection Points

Solving \(|x+2|-5=2\) requires considering two cases due to the absolute value.**Case 1**: \(x+2-5=2\)\(x-3=2\)\(x=5\)**Case 2**: \(-(x+2)-5=2\)\(-x-2-5=2\)\(-x-7=2\)\(-x=9\)\(x=-9\).Thus, the points of intersection are \(x=-9\) and \(x=5\).

Step 3: Graph Functions in Calculator

Enter the functions into your graphing calculator:- \(Y_1 = |x+2|-5\)- \(Y_2 = 2\)Plot the graphs to visualize the intersection points at \(x=-9\) and \(x=5\).

Step 4: Determine Shading Region

To find where \(|x+2|-5 < 2\), identify the region between \(x=-9\) and \(x=5\) where the graph of \(|x+2|-5\) is below the graph of \(y=2\). Shade this section on the x-axis.

Step 5: Write the Solution in Interval Notation

The solution to the inequality, represented in interval notation, describes the x-values where the graph of \(|x+2|-5\) is below the graph of \(y=2\). Therefore, the solution is the interval \((-9, 5)\).

Key Concepts

Absolute Value Inequalities

Absolute value inequalities, like the one given by \(|x+2|-5<2\), are expressions where an absolute value is set against an inequality sign. The absolute value, \(|x|\), represents the distance of a number \(x\) from zero on the number line. Because distance is always non-negative, absolute value equations consider two scenarios:

• The expression inside the absolute value is positive.
• The expression inside the absolute value is negative.

For the given inequality, we start by transforming it into the equivalent equations by treating it as an equality: \(|x+2|-5=2\). From here, we determine the roots of the equation. Solving these helps in finding the boundaries for values satisfying the inequality. Remember, absolute value splits the problem into two cases: one for when the expression inside is positive, and one for when it's negative. These cases give us intersection points where the graphs can be visualized and understood more clearly in subsequent steps.

Graphing Inequalities

Graphing inequalities involves plotting functions on a graph to visually identify the solution area. For the inequality \(|x+2|-5<2\), you graph two functions:

• \(Y_1 = |x+2|-5\)
• \(Y_2 = 2\)

By using a graphing calculator or any graphing tool, input these functions to find where the graph of \(|x+2|-5\) intersects and stays below the line \(y=2\). This visual approach makes it easier to understand where values of \(x\) satisfy the inequality. The shaded region between these intersection points is the solution set where the first graph is under the second graph, showing inequality satisfaction. Graphing not only helps in visualization but also provides a method to verify analytical solutions you've obtained.

Intersection Points

Intersection points are the x-values at which two graphs meet. In our scenario, for the inequality \(|x+2|-5<2\), solving the equations \(|x+2|-5=2\) gives the intersection points. By solving the absolute value equations, we found:

• \(x = 5\)
• \(x = -9\)

These points are essential because they define the boundary of regions where the inequality \(|x+2|-5<2\) is true. Intersection points give rise to critical values at which the behavior of the inequality changes — from true to false or vice versa. Therefore, these intersections help demarcate the interval within which the inequality holds true, informing how we express the solutions visually and in written form.

Interval Notation

Interval notation provides a concise way to express the set of solutions for inequalities. Once you have graphed and identified the intersection points, recognize the effective range for your inequality. Here, for the inequality \(|x+2|-5<2\), the graph shows that values of \(x\) between \(-9\) and \(5\) satisfy the inequality condition. The interval notation \((-9, 5)\) indicates that all points between \(-9\) and \(5\) are solutions but these endpoints are excluded. The parentheses signify that the boundary points are not part of the solution, aligning with the strict inequality \(<\) used in the original problem. Interval notation is an efficient way to communicate the solution set without relying solely on descriptive text or diagrams.