Problem 143 - 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. $$-2(x+4)>6 x+16$$

Step-by-Step Solution

Answer

The resulting solution is the x-values for which the graph of \( y=-2x-8\) lies above the graph of \( y=6x+16 \), as determined from the graph.

Step 1: Simplify each side

First, simplify each side of the inequality. The left side simplifies to \( -2x-8 \). The right side simplifies to \( 6x+16 \). So the inequality is now \( -2x-8 > 6x+16 \).

Step 2: Graph the functions

Now, graph the left side function, \( y = -2x-8 \), and the right side function, \( y = 6x+16 \), on the same graph. Note where the graph for the left function lies above the graph for the right function.

Step 3: Determine the solution

The solution to the inequality is the x-values for which the graph for the left side lies above the graph for the right side. These can be determined visually from the graph or by identifying the intersection point of the two lines and testing the values on both sides.

Key Concepts

Graphical Solution of Inequalities

When it comes to solving inequalities, the graphical method can be particularly intuitive. A graphical solution involves plotting the two sides of an inequality on a coordinate plane and determining the regions where one side exceeds the other.
The process of solving the inequality \( -2(x+4)>6x+16 \) graphically starts with simplification and then plotting the corresponding functions. For example, we simplify and plot \( y=-2x-8 \) and \( y=6x+16 \) on the same graph. The region where the line corresponding to \( y=-2x-8 \) lies above the line for \( y=6x+16 \) represents the solution set for the inequality. This visualization helps students see the solution rather than just calculating it, enhancing their understanding of the relationship between algebraic expressions and their graphical representation.

Finding the Intersection Point

One crucial aspect is the intersection point of the two lines, which defines the boundary of the solution set. By identifying this point, students can test values on each side to determine exactly which x-values satisfy the original inequality.

Linear Inequalities in One Variable

Linear inequalities in one variable, such as \( -2(x+4)>6x+16 \) after simplification, are solved much like linear equations but with the added complexity of a 'greater than' or 'less than' sign instead of an equality.
To solve these inequalities, one must first simplify the inequality to isolate the variable on one side. After simplifying, the inequality can be represented visually by graphing the linear functions that correspond to each side of the inequality. It is important to understand the direction of the inequality sign as it dictates which side of the graph is the solution region—where the inequality holds true. These concepts aid students in grasping not only the process of solving the inequality but also the meaning behind the solutions they find.

Inequality Simplification

Simplification is the first and one of the most essential steps in solving any inequality. It involves performing algebraic operations to isolate the variable, making the inequality easier to work with. For instance, in the inequality \( -2(x+4)>6x+16 \) simplification leads to \( -2x-8 > 6x+16 \) by distributing the -2 and bringing like terms together.
The goal is to transform the complex inequality into a more manageable form without altering the inequality's solution set. Care must be taken not to multiply or divide by negative numbers, as this reverses the inequality sign. By simplifying inequalities, students can then move on to solving them through various methods, including graphing, with a clearer representation of what they are working with.

Graphing Linear Functions

Graphing linear functions plays a pivotal role in visual learning and understanding of algebra. Linear functions, which are typically written in the form \( y=mx+b \) where \( m \) is the slope and \( b \) is the y-intercept, can represent one side of an inequality.
When graphing the functions derived from an inequality, like \( y=-2x-8 \) and \( y=6x+16 \) in the provided exercise, each line's slope and intercept create a visual representation of the equation. This visual representation allows for an easy comparison of the two sides of an inequality. When one line lies completely above or below another line, we can identify the solution set of the inequality. This graphical perspective reinforces the student’s understanding of not only the specific solution but the general behavior of linear functions.