Solve the nonlinear inequality. Express the solution using interval notation. [tex]$\frac{x+8}{2 x-1} \geq 0$[/tex]
Answer (2)
Find the critical points by setting the numerator and denominator to zero: x = − 8 and x = 2 1 .
Test the intervals ( − ∞ , − 8 ) , ( − 8 , 2 1 ) , and ( 2 1 , ∞ ) to determine where the inequality holds.
Include the point x = − 8 since the inequality is non-strict ( ≥ 0 ), but exclude x = 2 1 because the expression is undefined there.
Express the solution in interval notation: ( − ∞ , − 8 ] ∪ ( 2 1 , ∞ ) .
Explanation
Problem Analysis We are given the inequality 2 x − 1 x + 8 ≥ 0 . Our goal is to solve for x and express the solution in interval notation.
Finding Critical Points To solve this inequality, we first find the critical points by setting the numerator and denominator equal to zero:
Numerator: x + 8 = 0 ⟹ x = − 8
Denominator: 2 x − 1 = 0 ⟹ x = 2 1
Creating Intervals Now, we create a number line and mark the critical points − 8 and 2 1 . These points divide the number line into three intervals: ( − ∞ , − 8 ) , ( − 8 , 2 1 ) , and ( 2 1 , ∞ ) . We will test a point in each interval to determine where the inequality holds.
Testing Intervals
Interval ( − ∞ , − 8 ) : Choose x = − 9 . Then 0"> 2 x − 1 x + 8 = 2 ( − 9 ) − 1 − 9 + 8 = − 18 − 1 − 1 = − 19 − 1 = 19 1 > 0 . So, the interval ( − ∞ , − 8 ] is part of the solution.
Interval ( − 8 , 2 1 ) : Choose x = 0 . Then 2 x − 1 x + 8 = 2 ( 0 ) − 1 0 + 8 = − 1 8 = − 8 < 0 . So, the interval ( − 8 , 2 1 ) is not part of the solution.
Interval ( 2 1 , ∞ ) : Choose x = 1 . Then 0"> 2 x − 1 x + 8 = 2 ( 1 ) − 1 1 + 8 = 1 9 = 9 > 0 . So, the interval ( 2 1 , ∞ ) is part of the solution.
Final Solution Since the inequality is ≥ 0 , we include the point where the numerator is zero, x = − 8 . However, we exclude the point where the denominator is zero, x = 2 1 , because the expression is undefined at that point. Therefore, the solution in interval notation is ( − ∞ , − 8 ] ∪ ( 2 1 , ∞ ) .
Examples
Imagine you are analyzing the profit margin of a product. Let x represent the number of units sold. Suppose the profit margin is given by the expression 2 x − 1 x + 8 . You want to find the range of units you need to sell to ensure the profit margin is non-negative. Solving the inequality 2 x − 1 x + 8 ≥ 0 helps you determine the sales levels that guarantee a non-negative profit margin. This type of analysis is crucial for business planning and financial forecasting.
To solve the inequality 2 x − 1 x + 8 ≥ 0 , we identify critical points and test intervals. The solution in interval notation is ( − ∞ , − 8 ] ∪ ( 2 1 , + ∞ ) . This indicates where the expression is non-negative, including the point where it equals zero and excluding points where the expression is undefined.
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