Problem 142 - 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. \(-3(x-6)>2 x-2\)

Step-by-Step Solution

Answer

The values of \(x\) for which the graph for the left side lies above the graph for the right side (or -3(x-6) > 2x-2) is when \(x < \frac{18}{5}\).

Step 1: Simplify the inequality

First, distribute the '-3' to both 'x' and '-6' within the brackets on the left side of the inequality to get \(-3x+18>2x-2\). Then, isolate 'x' on one side of the inequality by adding '3x' to both sides and adding '2' to both sides. This gives: \(18 > 5x\) or, when we divide both sides by ‘5’, we get \(x < \frac{18}{5}\).

Step 2: Graph the inequality

Plot the two sides of the equation \(-3(x-6)\) and \(2x-2\), as separate graphs using a graphing utility (such as Desmos). The first graph will be a line passing through the point (6,-18), with a slope of -3. The second graph will be a line passing through the point (1,0), with a slope of 2.

Step 3: Identify where left graph is above right graph

The solution to our inequality \(x < \frac{18}{5}\) will be where the graph of \(-3(x-6)\) is above the graph of \(2x-2\). From the graph, this will be when x is less than \(\frac{18}{5}\).

Key Concepts

Graphing Utility

A graphing utility is a tool that helps to visually represent mathematical equations or inequalities. This can be particularly useful when solving linear inequalities such as \(-3(x-6)>2x-2\). By graphing each side of an inequality separately, you can easily determine where one graph is above the other, which helps in finding the solution.
There are various graphing utilities available, ranging from online platforms like Desmos to graphing calculators like the TI-84. These tools allow you to input equations and visually explore their graphs.

• They help to visualize complex problems that are not easy to solve algebraically.
• They provide an intuitive understanding of the relationship between equations.
• Some utilities offer additional functionalities like zooming, tracing, and finding intercepts.

Learning to effectively use a graphing utility is a crucial skill in mathematics, as it provides a dynamic way to solve and understand inequalities.

Linear Inequality

Linear inequalities are similar to linear equations, but instead of an equal sign \(=\), they use inequality signs \(>, <, \geq, \leq\). For example, \(-3(x-6)>2x-2\) is a linear inequality. Solving these requires finding the set of values for which the inequality holds true.
To solve a linear inequality, you often perform similar steps as solving equations, such as:

• Distributing coefficients like \(-3\) across terms inside parentheses.
• Isolating the variable on one side of the inequality by adding or subtracting terms from both sides.
• Dividing or multiplying both sides by a coefficient, remembering to reverse the inequality if multiplying or dividing by a negative number.

The main aim is to express the solution in terms of the variable, showing the range of values that satisfies the inequality.

Graphical Method

The graphical method involves plotting equations on a graph to find solutions to inequalities. In the case of \(-3(x-6)>2x-2\), each side of the inequality is represented as a line on a graph. For instance, you could plot \(-3(x-6)\) as one line and \(2x-2\) as another.
This approach is particularly useful for visual learners, as it provides a concrete representation of abstract algebraic concepts. By visually analyzing where the line representing the left side is above the line representing the right side, you can determine the solution for the inequality.

• The intersection point of the two lines often gives critical information about the solution.
• This method improves understanding of how changes in one graph affect the other.
• It helps in recognizing the solution region more vividly than purely algebraic methods.

Using the graphical method is an effective way to gain deeper insights into inequalities and other relational expressions.

Slope-Intercept Form

The slope-intercept form of a linear equation is an efficient way to express a line, written as \(y = mx + b\). Here, \(m\) represents the slope, and \(b\) is the intercept on the y-axis.
Understanding this form is key when graphing linear equations and inequalities. For instance, converting each side of the inequality \(-3(x-6)\) and \(2x-2\) into slope-intercept forms makes it easier to graph:

• The equation \(-3(x-6)\) expands and rearranges into the form \(y = -3x + 18\).
• The equation \(2x-2\) is already in the slope-intercept form \(y = 2x - 2\).

Using their slope-intercept forms, these equations can be directly plotted on a coordinate plane, and their slopes and intercepts become apparent. These features allow for a straightforward comparison between the graphs, aiding in solving inequalities efficiently.