In Exercises 85–90, solve each inequality and graph the solution set on a real number line. (x-6

step 1 Okay, we want to find a value to x for where this function here is greater than zero. And I hope

In Exercises 85–90, solve each inequality and graph the...

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of two functions, typically polynomials, set within an inequality sign. In solving these, one must determine where the rational expression is greater than, less than, or equal to zero. This requires a careful analysis of both the numerator and denominator and understanding how their signs affect the overall expression.

Domain Considerations

Before solving a rational inequality, it is important to identify the domain of the expression, particularly by determining the values that make the denominator zero. These values are excluded from the solution set because division by zero is undefined, and they act as critical boundaries when analyzing the behavior of the rational function.

Critical Points

Critical points are the values at which the numerator or the denominator of a rational expression becomes zero. These points divide the number line into distinct intervals where the sign of the expression may vary. In the context of rational inequalities, determining the critical points is essential for setting up interval tests to ascertain the intervals on which the inequality holds.

Sign Chart Analysis

A sign chart is a systematic method of determining the sign of a rational expression over various intervals defined by its critical points. By selecting test points within each interval, one can assess whether the entire expression is positive or negative. This analysis is crucial in identifying the solution set that satisfies the inequality.

Graphing on the Real Number Line

Graphing solutions on the real number line involves representing the intervals where the inequality is satisfied through open or closed circles and shading the appropriate regions. Open circles are used to indicate values that are not included in the solution set (such as points where the function is undefined or when the inequality is strict), while closed circles denote values that are included. This graphical representation provides a clear visual summary of the solution set.

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