Problem 143 - College Algebra

Question

Solve inequality using a graphing utility. Graph side separately. Then determine the values of \(x\) for which the graph for the left side lies above the graph for the right side. \(-2(x+4)>6 x+16\)

Step-by-Step Solution

Answer

Without knowing the specifics of the graph or the graphing utility used, a definitive solution can't be given. The solution would be the set of all \(x\) values for which the graph of \(-2x - 8\) is above the graph of \(6x + 16\).

Step 1: Simplify Both Sides of the Inequality

Begin by simplifying both sides of the inequality. For the left side, distribute the -2 to get \(-2x - 8\).

Step 2: Graph the Functions

Using a graphing utility, plot both \(-2x - 8\) and \(6x + 16\) on the same graph. Each function should be graphed separately to clearly distinguish which graph is above the other.

Step 3: Identify the Regions Where Left-Side Graph is Above the Right-Side Graph

Observe the graph and note the values of \(x\) where the graph of \(-2x - 8\) is above the graph of \(6x + 16\). These \(x\) values are the solutions to the inequality.

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial step in solving inequalities. By breaking down more complex expressions into simpler forms, we make it easier to work with them, especially when graphed.
Initially, in our exercise, we have the inequality \(-2(x+4) > 6x + 16\). The first step is to simplify both sides.

• Start by distributing the -2 on the left side of the inequality. This means you multiply -2 by each term inside the parenthesis.
• Thus, \(-2(x+4)\) simplifies to \(-2x - 8\).

Simplifying ensures that expressions are as compact as possible, which is essential when comparing them graphically. It helps in minimizing errors and enhances clarity, especially as we proceed to the graphing stage.

Graphing Utility

Using a graphing utility is very helpful to visualize equations and inequalities. It allows us to see where one graph stands in relation to another.
Graphing utilities can be calculators, software programs, or online tools that help plot equations quickly and accurately.

• In our example, we need to plot two separate equations: \(-2x - 8\) and \(6x + 16\).
• Make sure to enter these functions correctly in the graphing tool.

By plotting these lines, the graphing utility can visually show us which line is above the other, a visualization key to solving our inequality. These tools are great for tackling complex algebraic problems without manual plotting errors.

Algebraic Solutions

Algebraic solutions refer to solving mathematical expressions using algebra techniques. While the graph offers a visual solution, algebraic methods are essential for confirming findings.
For our inequality, simplifying and graphing provide a direction, but algebra helps us fully determine the solution.

• Begin with the inequality \(-2x - 8 > 6x + 16\).
• To solve algebraically, move all terms involving \(x\) to one side and constants to the other.
• This results in: \(-2x - 6x > 16 + 8\), simplifying further to \(-8x > 24\).
• Divide by -8 (remember: dividing by a negative flips the inequality): \(x < -3\).

The algebraic approach assures mathematical accuracy, verifying what is visually interpreted from the graph.

Linear Functions

Linear functions are equations that form straight lines when graphed. They are vital in understanding many algebraic problems, as they simplify the relationship between variables.
In any linear function, such as \(-2x - 8\) or \(6x + 16\), the highest exponent on the variable \(x\) is 1, ensuring it follows a straight line trajectory on a graph.

• Linear functions have the general form \(ax + b\); here, \(a\) is the slope, and \(b\) is the y-intercept.
• The slope determines the angle or steepness of the line, while the y-intercept shows where the line crosses the y-axis.

Understanding these components helps predict how changes in variables affect graph positioning, critical in inequality solutions like those tackled in this exercise. Grasping linear functions lays groundwork for more complex mathematical and real-world applications.