Problem 119 - 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 \(3.7 x-11.1=0 .\) How many solutions are there? How many solutions are there to any conditional linear equation in one variable?

Step-by-Step Solution

Answer

The solution is \( x = 3 \) with exactly one solution for this linear equation.

Step 1: Set the Equation to Find the x-intercept

To find the solution set of the equation using the x-intercept method, we start with the given equation: \[ 3.7x - 11.1 = 0 \]. To find the x-intercept, set this equation equal to zero.

Step 2: Solve for x

Solve the equation for \( x \):\[ 3.7x = 11.1 \].Now divide both sides by 3.7:\[ x = \frac{11.1}{3.7} \].Calculate the division to find \( x \):\[ x = 3 \].

Step 3: Interpret the Solution

The solution to the equation is \( x = 3 \). Since this is a linear equation in one variable, it will always have exactly one solution. Therefore, there are one solution in this case.

Key Concepts

Solution Set

When solving a linear equation like \(3.7x - 11.1 = 0\), we refer to its 'Solution Set'. This is simply the collection of all possible values of \(x\) that make the equation true. For linear equations with one variable, the solution set will always consist of a single value. In other words, the equation will have a unique solution.

A linear equation in the form \(ax + b = 0\) (\(a eq 0\)) can be solved by isolating \(x\). Once isolated, the unique value is the complete solution set of the equation. It's essential to verify this single value solution aligns with the structure of a conditional linear equation. Conditional linear equations assert that they hold true for only one specific value of \(x\), unlike identities, which hold for all values of \(x\).

X-intercept Method

The X-intercept Method is a popular technique to find solutions of linear equations. Here, you'll determine where the graph of the equation crosses the x-axis, essentially solving for \(x\) when the equation equals zero.

For the equation \(3.7x - 11.1 = 0\), the first step is to set the equation to equal zero, as mentioned. Solving for \(x\) means isolating \(x\) on one side of the equation. If there are coefficients or constants with \(x\), like 3.7 in this problem, you'll undo them using inverse operations, like division in this instance, as follows:

• Set \(3.7x = 11.1\)
• Divide both sides by 3.7 to find \(x\)
• Calculate \(x = \frac{11.1}{3.7}\)

This calculation reveals \(x = 3\), which tells us that the graph intersects the x-axis at \(x = 3\). This is the single solution to the equation, hence the name X-intercept Method.

Interval Notation

Interval Notation is a mathematical shorthand used to express solution sets over a range of values. However, in the context of a linear equation with one variable such as \(3.7x - 11.1 = 0\), the solution is not a range but a single point.

Unlike inequalities, where intervals are used to show a span of values that satisfy the comparison, the solution set of this equation is a specific number: \(3\). For single values, you would simply note the solution as \(\{3\}\) rather than using interval notation, which often applies to inequalities.

To summarize, interval notation is usually more relevant when exploring ranges in inequalities, but knowing it helps in comprehending the broader landscape of mathematical solutions where one often maneuvers between specific points and broad ranges.