Problem 77 - Algebra and Trigonometry

Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the horizontal asymptote of $$ h(x)=\frac{-3 x^{2}-7 x+1}{4-9 x^{2}} $$

Step-by-Step Solution

Answer

The solution is \left( -\infty, \frac{2}{5} \right] \cup \left[ \frac{4}{3}, +\infty \right).

Step 1: Understand the Absolute Value Inequality

The given inequality is \(|4x - 3| \geq x + 1 \). Absolute value inequalities of the form \(|A| \geq B \) are solved by considering the two scenarios: \( A \geq B \) and \( A \leq -B \).

Step 2: Set Up the Two Inequalities

Rewrite \( |4x - 3| \geq x + 1 \) as two separate inequalities: \( 4x - 3 \geq x + 1 \) and \( 4x - 3 \leq -(x + 1) \).

Step 3: Solve the First Inequality

Solve \( 4x - 3 \geq x + 1 \): \( 4x - 3 \geq x + 1 \ 4x - x \geq 1 + 3 \ 3x \geq 4 \ x \geq \frac{4}{3} \)

Step 4: Solve the Second Inequality

Solve \( 4x - 3 \leq -(x + 1) \): \( 4x - 3 \leq -x - 1 \ 4x + x \leq -1 + 3 \ 5x \leq 2 \ x \leq \frac{2}{5} \)

Step 5: Combine the Solutions

The solution to the inequality is the union of the solutions of the two inequalities found in Steps 3 and 4: \( x \leq \frac{2}{5} \) OR \( x \geq \frac{4}{3} \ The solution set is \left( -\infty, \frac{2}{5} \right] \cup \left[ \frac{4}{3}, +\infty \right) \).

Key Concepts

inequalities

Inequalities are mathematical expressions that describe the relative size or order of two values. Unlike equations, which use an equal sign, inequalities use symbols like \( > \), \( < \), \( \geq \), and \( \leq \). These symbols show that one side of the inequality is either greater than, less than, greater than or equal to, or less than or equal to the other side. Inequalities are used in various fields to compare quantities and establish ranges.
Understanding how to manipulate and solve inequalities is a key skill in algebra. It allows you to find ranges of values that satisfy certain conditions from equations involving variables.

absolute value

The absolute value of a number is its distance from zero on the number line, regardless of direction. For example, the absolute value of both \( -5 \) and \( 5 \) is \( 5 \). It's represented as \( |x| \).
In the context of equations and inequalities, absolute value expressions can represent two scenarios because the value inside the absolute value could be positive or negative. Thus, to solve \(|A| \geq B \), you need to split it into two separate inequalities: \( A \geq B \) and \( A \leq -B \). This provides a broader solution set that accounts for the dual nature of absolute values.

solving inequalities

Solving inequalities involves finding the values of the variable that make the inequality true. This is similar to solving equations but with special rules for direction and combining results.
For example, to solve \( 4x - 3 \geq x + 1 \), you first isolate \( x \) by subtracting \( x \) from both sides, giving \( 3x \geq 4 \). Dividing by \( 3 \) results in \( x \geq \frac{4}{3} \). Special care is required if you multiply or divide both sides of the inequality by a negative number, as this reverses the inequality sign.
When dealing with absolute values, after splitting the inequality into two parts (such as \( 4x - 3 \leq -x - 1 \) leading to \( 5x \leq 2 \) and thus \( x \leq \frac{2}{5} \)), you find the union of the resulting ranges.

union of solution sets

In the final step of solving an absolute value inequality, you often need to combine solution sets. This involves taking the individual solutions from the two inequalities and merging them.
For example, from the inequalities \( x \leq \frac{2}{5} \) and \( x \geq \frac{4}{3} \), the union of the solutions means that any value of \( x \) that satisfies at least one of these inequalities is part of the overall solution. The union of these solution sets is written as \( (-\infty, \frac{2}{5}] \cup [ \frac{4}{3}, +\infty ) \). This range captures all possible values of \( x \) that meet either inequality.