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Example: Solving a Rational Inequality from a Function
To determine when a rational function is less than or equal to a specific value, we can set up and solve a rational inequality. For example, given the function , to find the values of that make the function less than or equal to , we solve the inequality . First, find the zero partition numbers by setting the numerator and denominator to zero: and . These partition numbers divide the number line into three intervals: , , and . Testing values in each interval reveals that the quotient is negative in the interval . Since the inequality requires the expression to be less than or equal to , the solution must include the values where the quotient is negative or zero. The partition number must be excluded because it makes the denominator zero (and thus the function undefined), while is included because it makes the numerator, and the entire function, equal to . Therefore, written in interval notation, the solution is .
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Ch.7 Rational Expressions and Functions - Intermediate Algebra @ OpenStax
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Example: Solving a Rational Inequality from a Function
Example: Solving a Rational Inequality for Average Cost
As a supply chain logistics trainee, you are documenting the standard procedure for determining production ranges where the cost-to-revenue ratio falls below a specific threshold. This optimization model requires solving rational inequalities. Arrange the mathematical steps for solving a rational inequality into the correct procedural order for your algorithm documentation.
In your role as a technical analyst documenting standard operating procedures, you are outlining the steps for solving rational inequalities used in resource allocation models. Match each procedural step with its primary mathematical objective as defined in the standard systematic method.
As a laboratory assistant documenting the standard operating procedure for calculating concentration safety thresholds, you are outlining the method for solving rational inequalities. According to the systematic procedure, what is the mandatory first step you must perform before identifying any critical values or test intervals?
In your role as a technical analyst documenting safety compliance ranges, you must adhere to the systematic procedure for solving rational inequalities. True or False: According to this procedure, any value that makes the denominator of a rational expression equal to must always be excluded from the final solution set, regardless of whether the inequality uses non-strict symbols like or .
Procedural Step for Sign Evaluation in Rational Inequalities
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A logistics supervisor uses the rational function to determine optimal shipping routes based on cost-efficiency. When solving the inequality to find the range of optimal production levels, the supervisor identifies as a partition number. Based on the standard procedure for solving rational inequalities, why must the value 50 be excluded from the final solution set?
A production supervisor uses the rational function to monitor daily unit variance . To find the variance levels where the efficiency rating is non-positive (), the supervisor follows the standard rational inequality procedure. Arrange the following steps in the correct order to arrive at the solution .
An operations analyst is using the rational function to determine when a production line's waste-to-output ratio is within an acceptable range, expressed as . True or False: In the final solution set for this inequality, the value 5 should be included (using a square bracket) because the inequality symbol includes the 'or equal to' () condition.
A manufacturing supervisor uses the rational function to monitor daily production variance. To find the variance levels where the efficiency index is non-positive, the supervisor solves the inequality . Match each component of this mathematical procedure with its correct description.
Interpreting Rational Inequalities in a Logistics Context