Solve the nonlinear inequality. Express the solution using interval notation.
[tex]$\frac{x+8}{3 x-1} \geq 0$[/tex]
Answer (2)
Find the critical points of the inequality by setting the numerator and denominator to zero: x = − 8 and x = 3 1 .
Test intervals ( − ∞ , − 8 ) , ( − 8 , 3 1 ) , and ( 3 1 , ∞ ) to determine where the expression is positive or negative.
Include intervals where the expression is non-negative, and include x = − 8 but exclude x = 3 1 .
Express the solution in interval notation: ( − ∞ , − 8 ] ∪ ( 3 1 , ∞ ) .
Explanation
Analyze the Inequality We are given the inequality 3 x − 1 x + 8 ≥ 0 . To solve this inequality, we need to find the intervals where the expression is non-negative. First, we find the critical points by setting the numerator and denominator equal to zero.
Find Critical Points The numerator is zero when x + 8 = 0 , which gives x = − 8 . The denominator is zero when 3 x − 1 = 0 , which gives x = 3 1 . These critical points divide the real number line into three intervals: ( − ∞ , − 8 ) , ( − 8 , 3 1 ) , and ( 3 1 , ∞ ) .
Test Intervals Now, we test a value from each interval to determine the sign of the expression 3 x − 1 x + 8 in that interval.
Interval ( − ∞ , − 8 ) : Let x = − 9 . Then 0"> 3 ( − 9 ) − 1 − 9 + 8 = − 28 − 1 = 28 1 > 0 . So the expression is positive in this interval.
Interval ( − 8 , 3 1 ) : Let x = 0 . Then 3 ( 0 ) − 1 0 + 8 = − 1 8 = − 8 < 0 . So the expression is negative in this interval.
Interval ( 3 1 , ∞ ) : Let x = 1 . Then 0"> 3 ( 1 ) − 1 1 + 8 = 2 9 > 0 . So the expression is positive in this interval.
Determine Solution Intervals Since we want the expression to be greater than or equal to zero, we include the intervals where the expression is positive. Also, we include the point where the numerator is zero ( x = − 8 ) because the inequality is non-strict ( ≥ 0 ). However, we do not include the point where the denominator is zero ( x = 3 1 ) because the expression is undefined at this point.
State the Solution Therefore, the solution to the inequality is ( − ∞ , − 8 ] ∪ ( 3 1 , ∞ ) .
Examples
Understanding inequalities like this is crucial in many real-world scenarios. For instance, in business, you might use them to determine the range of sales needed to ensure a profit, considering fixed costs and variable revenues. Similarly, in physics, inequalities can help define the conditions under which a certain phenomenon occurs, such as the range of angles for a projectile to reach a target. In everyday life, you might use them to manage your budget, ensuring your expenses stay within your income. Inequalities provide a powerful tool for making informed decisions and managing constraints effectively.
The solution to the inequality 3 x − 1 x + 8 ≥ 0 is given by the intervals ( − ∞ , − 8 ] ∪ ( 3 1 , ∞ ) . This means the expression is non-negative for all x values in these intervals. The critical points for this inequality were found by setting the numerator and denominator to zero and testing the intervals they create.
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