Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation. [tex]$\frac{x+1}{x+8}\ \textless \ 2$[/tex]
Answer (2)
The solution to the rational inequality x + 8 x + 1 < 2 is ( − ∞ , − 15 ) ∪ ( − 8 , ∞ ) . This means the values of x can be less than -15 or greater than -8. Both -15 and -8 are not included in the solution set.
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Subtract 2 from both sides: x + 8 x + 1 − 2 < 0 .
Combine terms: x + 8 − x − 15 < 0 .
Multiply by -1: 0"> x + 8 x + 15 > 0 .
Find critical values and test intervals: The solution is x < − 15 or -8"> x > − 8 , which in interval notation is ( − ∞ , − 15 ) ∪ ( − 8 , ∞ ) .
Explanation
Problem Analysis We are asked to solve the rational inequality x + 8 x + 1 < 2 and express the solution set in interval notation. We also need to graph the solution set on a real number line.
Subtracting 2 First, we subtract 2 from both sides of the inequality to get x + 8 x + 1 − 2 < 0 .
Combining Terms Next, we find a common denominator and combine the terms on the left side: x + 8 x + 1 − 2 ( x + 8 ) < 0 x + 8 x + 1 − 2 x − 16 < 0 x + 8 − x − 15 < 0
Multiplying by -1 Now, we multiply both sides by -1, remembering to flip the inequality sign: 0"> x + 8 x + 15 > 0
Finding Critical Values We find the critical values by setting the numerator and denominator equal to zero: x + 15 = 0 ⇒ x = − 15 x + 8 = 0 ⇒ x = − 8
Creating a Sign Chart We create a sign chart using the critical values -15 and -8 to determine the intervals where the expression x + 8 x + 15 is positive. The intervals are ( − ∞ , − 15 ) , ( − 15 , − 8 ) , and ( − 8 , ∞ ) .
Testing Intervals We test values in each interval to determine the sign of the expression. For x < − 15 , test x = − 16 : 0"> − 16 + 8 − 16 + 15 = − 8 − 1 = 8 1 > 0 For − 15 < x < − 8 , test x = − 10 : − 10 + 8 − 10 + 15 = − 2 5 < 0 For -8"> x > − 8 , test x = 0 : 0"> 0 + 8 0 + 15 = 8 15 > 0
Solution Set The expression x + 8 x + 15 is positive when x < − 15 or -8"> x > − 8 . Therefore, the solution set in interval notation is ( − ∞ , − 15 ) ∪ ( − 8 , ∞ ) .
Final Answer The solution set is ( − ∞ , − 15 ) ∪ ( − 8 , ∞ ) .
Examples
Rational inequalities are useful in various real-world scenarios, such as determining the safe operating range for equipment or calculating the optimal levels for chemical mixtures. For example, suppose a chemical reaction requires the concentration of a reactant to be within a certain range to prevent unwanted side effects. If the concentration is given by a rational function of time, solving a rational inequality can help determine the time intervals during which the reaction proceeds safely and efficiently. Understanding and solving rational inequalities allows engineers and scientists to make informed decisions and maintain control over complex processes.
