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Example: Solving a Rational Inequality Using Interval Notation
To solve the rational inequality , first identify the zero partition numbers where the numerator or denominator equals zero. The numerator when , and the denominator when . These zero partition numbers, and , divide the number line into three intervals: , , and . Testing a value in each interval—such as , , and —reveals the sign of the quotient. The quotient is positive in and , and negative in . Because the inequality requires the expression to be greater than or equal to zero, the intervals and are solutions. The partition number must be excluded with a parenthesis because it makes the denominator zero, yielding an undefined expression. The partition number is included with a bracket because it makes the rational expression zero, which satisfies the condition. Connecting the two valid intervals with the union symbol gives the final solution: .
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Ch.7 Rational Expressions and Functions - Intermediate Algebra @ OpenStax
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Example: Solving a Rational Inequality Using Interval Notation
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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.
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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 technical analyst is solving the rational inequality to determine an operational threshold. During the process, the analyst identifies the partition numbers 1 and -3. Why is the value -3 excluded from the final solution set using a parenthesis, while the value 1 is included using a bracket?
A business analyst is using the rational inequality to determine the profitability range of a new product line. Match each component of the solution process with its correct role or characteristic.
A logistics coordinator is following a standardized protocol to solve the rational inequality to determine an 'Efficiency Index' for a warehouse operation. Arrange the following steps in the correct order based on the solving procedure provided in the example.
Load Balance Operational Range
A logistics coordinator is solving the rational inequality to determine an 'Efficiency Index' for a shipping route. The values 1 and -3, which are identified to divide the number line into intervals for sign testing, are referred to as zero __________ numbers.