Problem 142 - College Algebra Essentials

Question

Solve each inequality using a graphing utility. Graph each side separately. Then determine the values of x for which the graph for the left side lies above the graph for the right side. $$-3(x-6)>2 x-2$$

Step-by-Step Solution

Answer

The inequality \(-3(x-6) > 2x - 2\) is solved when \(x > 4\).

Step 1: Rewrite the inequality

First, simplify the inequality to a form that is easier to understand and plot. Distribute the \(-3\) to the expression inside the parentheses on the left side: \(-3x + 18 > 2x - 2\). Then, gather terms with \(x\) on one side and the constants on the other side to get: \(-5x > -20\).

Step 2: Graph the inequality

Using a graphing utility, plot the two functions \(y=-5x\) and \(y=-20\). The solution to the inequality will be the regions where the graph of \(y=-5x\) (left side) is above the graph of \(y=-20\) (right side).

Step 3: Determine the solution

By observing the graph, the graph of \(y=-5x\) intersects the graph of \(y=-20\) when \(x=4\). For values greater than \(4\), the graph for the left side lies above that of the right side. Therefore, the solution to the inequality is \(x > 4\).

Key Concepts

Graphing Utilities

Graphing utilities are powerful tools that help visualize mathematical equations and inequalities. They are more than just graphing calculators; they can be software or online platforms designed to make graph plotting easier. By providing an interface to input equations, graphing utilities can:

• Quickly display complex graphs.
• Allow users to zoom in and out of graphs for better analysis.
• Help compare multiple graphs simultaneously.

In the context of solving inequalities, these utilities enable you to see where one graph stands relative to another. This visual representation is crucial for understanding how different parts of the inequality interact, and finding where the solutions exist. You get instant feedback on adjustments, making it easier to comprehend complex concepts.

Solving Inequalities

Solving inequalities is about finding the range of values that satisfy a particular mathematical statement. Unlike equations, which rely on equality, inequalities show a range of possibilities through symbols such as ">", "<", ">=" and "<=". When tackling inequalities:

• Understand that the goal is to isolate the variable on one side.
• Ensure you reverse the inequality symbol when multiplying or dividing by a negative number.
• Translate the inequality into a mathematical representation that's easy to compare, like a graph.

For instance, when you start with \(-3(x-6)>2x-2\), the focus is on manipulating this expression to highlight where the left side exceeds the right. Through conversion and simplification, the inequality transforms into a format that graphical methods can exploit for solutions.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying mathematical equations or inequalities to derive a solution. It involves performing operations such as addition, subtraction, multiplication, or division to both sides of the equation to isolate the variable. For the given inequality, \(-3(x-6)>2x-2\):

• First, distribute the \(-3\) across \((x-6)\) resulting in \(-3x + 18 > 2x - 2\).
• Next, bring terms involving \(x\) together by subtracting \(2x\) from both sides.
• Rearrange constants by moving the constant from the right to the left side to simplify to \(-5x > -20\).
• Finally, divide by \(-5\) and reverse the inequality to get \(x > 4\).

This strategy provides a clear path to solve the inequality, making it easier to understand and verify with graphical solutions.

Graphical Solution Methods

Graphical solution methods leverage visual elements to interpret and solve mathematical inequalities. This approach involves plotting each side of the inequality as a separate line or curve on a coordinate plane and analyzing their intersections and relative positions. Let's break down the steps for our inequality:

• Plot \(y = -5x\) and \(y = -20\), which arise from the simplified inequality.
• On a graph, observe these two lines and find their point of intersection.
• The line \(y = -5x\) will be above \(y = -20\) when \(x\) is greater than \(4\).

This visual approach not only supports the algebraic solution but also confirms it by showing graphically where one function exceeds the other. By examining the graph, you can directly see how the inequality \(-3(x-6)>2x-2\) eresolves into \(x > 4\), thus providing clarity and confirmation that the algebraic solution is correct.