Problem 121 - A Graphical Approach to Precalculus with Limits

Question

The solution set of a linear equation is closely related to the solution set of a linear inequality.In order to investigate this connection. Write answers in interval notation when appropriate. Use the \(x\) -intercept method to find the solution set of the equation $$-4 x+6=0$$ Then find the solution sets of the inequalities $$-4 x+6<0 \text { and }-4 x+6>0$$

Step-by-Step Solution

Answer

\(x = \frac{3}{2}\); \(x > \frac{3}{2}\) as \((\frac{3}{2}, \infty)\); \(x < \frac{3}{2}\) as \((-\infty, \frac{3}{2})\).

Step 1: Solve the Linear Equation

To solve the linear equation \(-4x + 6 = 0\), first isolate \(x\) by subtracting 6 from both sides: \(-4x = -6\). Then divide both sides by \(-4\) to find \(x\): \(x = \frac{-6}{-4} = \frac{3}{2}\). Therefore, the solution to the equation is \(x = \frac{3}{2}\).

Step 2: Solve the Inequality \(-4x + 6 < 0\)

First, subtract 6 from both sides to get \(-4x < -6\). Next, divide both sides by \(-4\) (remember to reverse the inequality sign when dividing by a negative number): \(x > \frac{3}{2}\). The solution in interval notation is \((\frac{3}{2}, \infty)\).

Step 3: Solve the Inequality \(-4x + 6 > 0\)

Again, subtract 6 from both sides: \(-4x > -6\). Divide both sides by \(-4\), reversing the inequality sign: \(x < \frac{3}{2}\). The solution in interval notation is \((-\infty, \frac{3}{2})\).

Step 4: Compare Solution Sets

The solution to the equation \(-4x + 6 = 0\) is a single point \(x = \frac{3}{2}\). The inequality \(-4x + 6 < 0\) has a solution set of \((\frac{3}{2}, \infty)\), and \(-4x + 6 > 0\) has a solution set of \((-\infty, \frac{3}{2})\). These intervals show the points where \(x\) makes each inequality true.

Key Concepts

Inequalities

Inequalities are mathematical expressions that represent the relationship between two values when they are not equal, often using symbols such as ">", "<", "≥", and "≤". Unlike equations, which demonstrate equality, inequalities show that one side is either larger or smaller. For example, an inequality expresses that one quantity is less than another.
In the context of the exercise, we explore inequalities with the expressions

• -4x + 6 < 0
• -4x + 6 > 0

Solving these inequalities involves transforming the inequality into a simpler form where the variable can be isolated. Be wary when dividing by a negative number since it reverses the inequality sign, showing just how a small change in the approach can lead to different solution sets.

X-Intercept Method

The x-intercept method is a technique used to find where a given linear equation crosses the x-axis. This occurs when the y-value of an equation is zero. Essentially, it involves setting the equation equal to zero and solving for the variable, typically x.
In the example

• -4x + 6 = 0

the x-intercept is determined by rearranging the equation and isolating x. This method reveals the precise point, \( x = \frac{3}{2} \), where the line meets the x-axis.

• It's a useful approach for quickly identifying solutions or aiding in graphing

as it swiftly highlights where changes are usually necessary to satisfy an inequality.

Solution Set

A solution set encompasses all the possible values of a variable that satisfy an equation or inequality. For a linear equation, the solution set might consist of a single value, whereas for inequalities, it typically involves a range or interval of values.
In the given linear equation

• -4x + 6 = 0, the solution set is the single value: \( x = \frac{3}{2} \)

On the other hand, for the inequality

• -4x + 6 < 0, the solution set is all values greater than \( \frac{3}{2} \) leading to

\( (\frac{3}{2}, \infty) \)
Alternatively, for

• -4x + 6 > 0, the solution set includes all values less than \( \frac{3}{2} \) yielding

\( (-\infty, \frac{3}{2}) \). By exploring different solution sets, you can understand the full scope of variable values that solve an inequality or equation.

Interval Notation

Interval notation is a simple way to represent a range of values within an inequality. It summarises the numbers that satisfy the inequality without listing them explicitly. For example, using parentheses and brackets, interval notation helps specify whether the endpoints are included or not.
A parenthesis,

• ( ), indicates that endpoints are not included, often used for strict inequalities like "<" or ">

." Alternatively

• , brackets, [ ] are used when endpoints are included, suitable for non-strict inequalities "≤" or "≥"

.In the exercise, using interval notation enabled us to express the solution sets \( (\frac{3}{2}, \infty) \) and

• (-\infty, \frac{3}{2})

effectively, offering an elegant and concise representation while understanding the range of solutions clearly. This notation is widely appreciated for its ability to neatly capture and convey information about inequalities.