Problem 142 - Algebra and Trigonometry

Question

In Exercises 142–143, 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

By observing the graph, it will be found that the graph of the equation y1 = -3x + 18 is above the graph of the equation y2 = 2x - 2 for values of x that are less than 4. Therefore, the solution to the inequality is \(x<4\).

Step 1: Simplify Each Side

To make it easier to solve, simplify the left and right sides of the inequality separately. For the left side, distribute the -3 to get \(-3x+18\). For the right side, simply rewrite it as \(2x - 2\). The simplified inequality is \(-3x + 18 > 2x - 2\).

Step 2: Graph each side separately

Use a graphing utility to graph each side of the inequality as individual functions. Graph the function y1 = -3x + 18 and the function y2 = 2x - 2.

Step 3: Determine where y1 is above y2

Examine the graph and see where the graph of the first equation (y1) is above the graph of the second equation (y2). This region corresponds to where \(-3x+18 > 2x-2\).

Step 4: Identify the interval of x values

Visually inspect which x-values on the graph place the y1 function above the y2 function. This interval of x values is the solution to the inequality.

Key Concepts

Graphing Utility

A graphing utility is a handy tool used to visualize mathematical functions easily. It can be software, an app, or even a graphing calculator. These utilities allow you to input equations and instantly plot their corresponding graphs.

• They help in understanding relationships between variables by showing graphical representations.
• You can simultaneously plot multiple functions to compare them visually.
• Many graphing utilities provide additional features like zooming, tracing points, and finding intersections.

In the context of the original exercise, using a graphing utility lets you graph each side of the given inequality as separate functions. This visual comparison is crucial in finding the solution set, as you can clearly see where one graph lies above or below another. The ability to visualize these intersections and overlaps is what makes graphing utilities indispensable when solving complex inequalities.

Linear Inequalities

Linear inequalities are expressions involving linear functions where a certain relationship, like greater than or less than, is stated between them. A linear inequality looks like a linear equation but with inequality signs such as <, >, ≤, or ≥ instead of an equal sign.

Examples of linear inequalities include:

• a > b + c
• 2x + 3 < 5x - 7

To graph linear inequalities, you generally graph the line as if it were an equation and then determine which side of the line satisfies the inequality by testing points. For the inequality provided in the exercise, after simplifying, you observe the linear functions on either side of the inequality and represent them graphically. Understanding these linear inequalities is foundational, as it leads to determining where one function is greater or less than the other.

Solving Inequalities

Solving inequalities involves finding the values of variables that make the inequality true. Unlike solving equations, solving inequalities often requires more attention since the inequality sign can change under certain operations.

Here are basic steps to solve inequalities:

• Simplify both sides of the inequality if needed.
• Get all variable terms on one side and constant terms on the other.
• Remember that multiplying or dividing by a negative number reverses the inequality sign.

In the original exercise, solving the inequality egin{align*} -3x + 18 > 2x - 2 egin{align*} is achieved by simplifying and then rearranging terms:

• Add 3x to both sides, giving 18 > 5x - 2.
• Add 2 to both sides, leading to 20 > 5x.
• Divide by 5 to conclude 4 > x or equivalently, x < 4.

This process converts the inequality into a clear range of possible x-values.

Inequality Solution Interval

An inequality solution interval gives all the possible values for which an inequality holds true. It describes the section of the number line where the inequality's conditions are satisfied.

These intervals can be expressed in various methods:

• Inequality notation, such as x < 4.
• Interval notation, written as (-∞, 4), which includes all numbers less than 4.
• Graphical form, using a number line to visually represent the set of solutions.

For the inequality from our exercise, the solution interval is where y1, the graph of -3x + 18, is above y2, the graph of 2x - 2. Solving visually or analytically gives x < 4. In interval notation, this solution is written as (-∞, 4), indicating all numbers less than 4 are solutions. This method helps in understanding both isolated and continuous solution zones.