Problem 42 - Calculus with Concepts in Calculus

Question

Solve the inequality. $$ \frac{2-5 x}{3-4 x} \geq-2 $$

Step-by-Step Solution

Answer

\( x \leq \frac{8}{13} \)

Step 1: Understand the Given Inequality

The original inequality given is \( \frac{2-5x}{3-4x} \geq -2 \). Our goal is to find the values of \( x \) that make this inequality true.

Step 2: Rearrange the Inequality

To solve an inequality involving a rational expression, move all terms to one side to make a single rational expression. This becomes \( \frac{2-5x}{3-4x} + 2 \geq 0 \).

Step 3: Combine Terms

The next step is to combine the terms into a single fraction. Find a common denominator, which is \( 3 - 4x \), to write the expression as \( \frac{2-5x + 2(3-4x)}{3-4x} \geq 0 \).

Step 4: Simplify the Numerator

Now simplify the numerator: \( 2 - 5x + 6 - 8x = 8 - 13x \). Thus, the inequality becomes \( \frac{8 - 13x}{3-4x} \geq 0 \).

Step 5: Find Critical Points

To solve \( \frac{8 - 13x}{3-4x} \geq 0 \), identify where the expression is equal to zero or undefined. Set the numerator equal to zero: \( 8 - 13x = 0 \) gives \( x = \frac{8}{13} \). Set the denominator equal to zero: \( 3-4x = 0 \) gives \( x = \frac{3}{4} \).

Step 6: Test Intervals

Divide the number line into intervals based on \( x = \frac{8}{13} \) and \( x = \frac{3}{4} \). Test points in each interval to determine where the inequality is satisfied. Check intervals: \( x < \frac{8}{13} \), \( \frac{8}{13} < x < \frac{3}{4} \), and \( x > \frac{3}{4} \).

Step 7: Analyze Each Interval

1. For \( x < \frac{8}{13} \), choose \( x = 0 \). The inequality becomes \( \frac{8}{3} > 0 \), which is true. 2. For \( \frac{8}{13} < x < \frac{3}{4} \), choose \( x = \frac{7}{10} \). The inequality is false. 3. For \( x > \frac{3}{4} \), choose \( x = 1 \). The inequality is true if the result is \( < 0 \), which for this test point becomes false.

Step 8: Formulate Solution Set

The solution set is where our inequality holds, which is for \( x \leq \frac{8}{13} \). At \( x = \frac{8}{13} \), the inequality holds since it is equivalent to zero at this point.

Key Concepts

Critical Points

Critical points are crucial in solving rational inequalities as they indicate where the inequality can change its truth value. To find these points, you must consider two scenarios:

• Where the fraction's numerator equals zero.
• Where the fraction's denominator equals zero, because division by zero is undefined.

In the context of the inequality \( \frac{8 - 13x}{3-4x} \geq 0 \), finding these points involves setting up equations. The numerator is set to zero, leading to \( 8 - 13x = 0 \), solving gives us \( x = \frac{8}{13} \). This is where the fraction changes sign. Next, solve the denominator \( 3 - 4x = 0 \), which yields \( x = \frac{3}{4} \). When we have these critical points, they break the number line into regions that we can test for the validity of the inequality.

Interval Testing

Once you have the critical points, you can perform interval testing. This process involves checking test values from the regions divided by the critical points to see where the inequality holds true.Let's consider the inequality \( \frac{8 - 13x}{3-4x} \geq 0 \). After finding critical points at \( x = \frac{8}{13} \) and \( x = \frac{3}{4} \), the number line is split into three ranges:

• \( x < \frac{8}{13} \)
• \( \frac{8}{13} < x < \frac{3}{4} \)
• \( x > \frac{3}{4} \)

By selecting a number within each interval and substituting it into the inequality, you check whether it satisfies the condition. For example, choosing \( x = \frac{7}{10} \) in the interval \( \frac{8}{13} < x < \frac{3}{4} \), shows the inequality is not satisfied there. Testing helps isolate which intervals satisfy the inequality, leading to a solution set.

Simplification of Expressions

Simplifying expressions in rational inequalities is about combining terms effectively so you can analyze them. This simplification usually involves combining fractions over a common denominator and reducing terms whenever possible.For example, starting with the inequality \( \frac{2-5x}{3-4x} \geq -2 \), we initially rearrange it to \( \frac{2-5x}{3-4x} + 2 \geq 0 \). The task then is to write these terms as a single fraction. The common denominator here is \( 3-4x \). Once rewritten, the numerator becomes \( 2 - 5x + 6 - 8x \), simplifying to \( 8 - 13x \). This reduction simplifies the complex expression and makes it easier to identify critical points, facilitating the rest of the inequality-solving process.