Problem 91 - Precalculus Student Solutions Manual 5th

Question

Solve each rational inequality. Write each solution set in interval notation.4 $$9\frac{(5 x-3)^{3}}{(25-8 x)^{2}} \leq 0$$ $$\frac{(2 x-3)(3 x+8)}{(x-6)^{3}} \geq 0 \quad\( 92. \)\frac{(9 x-11)(2 x+7)}{(3 x-8)^{3}}>0$$.

Step-by-Step Solution

Answer

Solution for first inequality: Solution for second inequality: Solution for third inequality.

Step 1: Analyze the rational inequality

Given the inequality dsfsd Identify the numerators and denominators.

Step 2: Find Critical Points - Inequality 1

Find the values of x that make the numerator and denominator zero. These critical points will divide the number line into intervals. For the first inequality: * The numerator is zero when * The denominator is zero when

Step 3: Test Intervals - Inequality 1

Determine the sign of the rational expression in each interval by choosing a test point from each interval. Evaluate whether the expression is positive or negative.

Step 4: Combine Intervals - Inequality 1

Combine the intervals to write the solution set in interval notation where the inequality holds true.

Step 5: Repeat for Inequality 2

Repeat Steps 2 through 4 for the second inequality:

Step 6: Repeat for Inequality 3

Repeat Steps 2 through 4 for the third inequality:

Key Concepts

Interval Notation

When solving rational inequalities, it's important to express the solution in interval notation. Interval notation uses brackets and parentheses to describe sets of numbers on the number line.
To use interval notation:

• Use a square bracket \(([] or []\) to include an endpoint in the interval. For example, \( [2, 5] \) means all numbers from 2 to 5, including 2 and 5.
• Use a parenthesis \( () \) to exclude an endpoint. For example, \( (2, 5) \) means all numbers between 2 and 5, but not including 2 or 5.

Consider all intervals where the rational expression is positive or negative based on the inequality sign, and piece them together using proper interval notation.

Critical Points

Critical points are values of x that make either the numerator or denominator zero. These points are crucial as they divide the number line into different intervals.
To find critical points:

• Set the numerator equal to zero and solve for x.
• Set the denominator equal to zero and solve for x.

These x-values split the number line, helping in determining the behavior of the rational function in those split intervals.
For example, in the inequality \( \frac{{(9x-11)(2x+7)}}{{(3x-8)^3}} >0 \), solving \( 9x-11 = 0 \) and \( 2x+7 = 0 \) gives critical points from the numerator, and solving \( 3x-8=0 \) gives critical points from the denominator.

Numerator and Denominator Analysis

In rational inequalities, both the numerator and denominator affect the sign of the entire expression. Analyzing their signs individually allows us to predict their behavior.
Steps to analyze them:

• Identify the factors in the numerator and denominator that could change the sign of the expression.
• Examine each factor individually to understand where they are positive or negative.

For example, with \( \frac{{(2x-3)(3x+8)}}{{(x-6)^3}} \), solving the individual components tells us where each part changes sign:
\( 2x-3=0 \) at \( x=1.5 \),
\( 3x+8=0 \) at \( x=-2.67 \),
and \( x-6=0 \) at \( x=6 \). This information is vital for determining the function's sign over different intervals.

Sign Testing

Once critical points are identified, sign testing helps determine whether the rational expression is positive or negative in each interval they form.
To perform sign testing:

• Choose any test point within each interval created by the critical points.
• Plug the test point into the rational inequality and check the sign of the resulting value.

For instance, if we have intervals \( (-∞, -2.67) \), \((-2.67, 1.5) \), \( (1.5, 6) \), and \( (6, ∞) \) from previous analysis, pick test points such as x=-3, x=0, x=2, and x=7. Substitute these into the original inequality to ascertain the positive or negative nature of the expression.
Combining these results gives the final intervals where the inequality holds true.