Problem 75 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 73-76: Express, in interval form, the \(x\)-values such that \(y_{1}

Step-by-Step Solution

Answer

Solve the inequality and find the intervals where \( y_1 < y_2 \).

Step 1: Understand the Inequality

The exercise asks us to find the interval where a function or expression \( y_1 \) is less than \( y_2 \). Essentially, we want to identify the values of \( x \) for which this inequality holds true.

Step 2: Set up the Inequality

Set up the inequality \( y_1 < y_2 \). Depending on the specific functions or expressions for \( y_1 \) and \( y_2 \), we need to determine when the left side is less than the right side.

Step 3: Solve the Inequality

Solve \( y_1 < y_2 \) for \( x \). This might require algebraic manipulation such as factoring, expanding, or even graphing the functions to find the overlap of \( x \)-values satisfying the inequality.

Step 4: Identify Points of Intersection

Determine the points of intersection, if any, by solving \( y_1 = y_2 \). These points help define the boundaries of our solution interval.

Step 5: Determine Solution Interval

Analyze the sign of \( y_1 - y_2 \) between the points of intersection to determine where \( y_1 < y_2 \). This involves checking if the inequality holds in the intervals between and possibly outside the intersection points.

Step 6: Write the Interval

Express the solution in interval notation, denoting the \( x \)-values where \( y_1 < y_2 \). Be sure each interval starts and ends with the correct type of bracket (e.g., \( (a, b) \) or \( [a, b] \)).

Key Concepts

Interval Notation

Interval notation is a way of representing a range of values. It's used to express the set of all numbers between and including specific points. For instance, the interval \(a, b\) refers to all numbers greater than \a\ but less than \b\. The round brackets mean that the endpoints are not included; these are called open intervals.
One might also encounter closed intervals denoted by square brackets, as in \[a, b\]. This includes the endpoints, meaning the values at \a\ and \b\ are part of the solution. There's also a mix of open and closed intervals like \(a, b\] or \[a, b\).
Here’s a quick guideline for brackets in interval notation:

• \(a, b\) - Open interval; neither endpoint is included.
• \[a, b\] - Closed interval; both endpoints are included.
• \(a, b\] or \[a, b\) - Semi-open interval; one endpoint included.

Using interval notation makes it clear which \x\-values satisfy the inequality in question.

Points of Intersection

Points of intersection are crucial in solving inequalities involving functions. These points occur where two graphs meet, meaning the outputs are equal at these \(x\) values. To find these points, set the functions equal to each other, such as solving \y_1 = y_2\. This helps us understand the transition from one function being greater than the other and is key to determining the solution interval.
Once you find these intersections, plot or note them. They act as boundaries and divide the real number line into sections that can be analyzed.

• Check within each section to determine where the inequality holds.
• Use these points to delineate the intervals in your final solution.

This method is not only helpful but necessary since the intersections define shifts in the inequality.

Inequality Solutions

To solve an inequality, the steps are to first set up the inequality by arranging both functions or expressions, such as \(y_1 < y_2\). This establishes that we are looking for \(x\) values where this relationship holds. The inequality solution process might require several techniques:

• Factoring - Decompose expressions into simpler product forms.
• Expanding - Distribute products to simplify each side for comparison.
• Graphing - Analyze visually to find overlapping regions or critical points.

Analyzing the relationship within the intervals broken by points of intersection ensures accuracy. Remember, the regions where the inequality holds true are our answer sets. These solutions then need to be stated using proper interval notation.

Algebraic Manipulation

Algebraic manipulation is a core method used in solving inequalities. It's like a toolbox that lets you adjust expressions to reveal comparisons more clearly. Here are some common techniques involved:

• Combining Like Terms - Simplify expressions by grouping similar items.
• Isolating Variables - Rearrange terms to get the variable on one side of the equation.
• Simplifying Expressions - Reduce expressions to their simplest forms for easier comparison.

These techniques may be necessary in converting raw inequality forms into ones that are more readily analyzable. By refining the inequality through algebraic manipulation, clearer insights into the values of \(x\) are gained, making it easier to determine which intervals satisfy the problem requirements.