Learning Outcomes
- Solve rational inequalities using boundary value method.
Solving Rational Inequalities using Boundary Value Method
Any inequality that can be put into one of the following forms
f(x)>0,f(x)≥0,f(x)<0,orf(x)≤0f(x)>0,f(x)≥0,f(x)<0,orf(x)≤0, where ff is a rational function
is called rational inequality.
How To: Solve Rational Inequalities
- Find restriction(s) of the given rational expression(s) in the inequality. The restriction(s) is(are) the boundary point(s).
- Rewrite the given rational inequality as an equation by replacing the inequality symbol with the equal sign.
- Solve the rational equation. The real solution(s) of the equation is(are) the boundary point(s).
- Plot the boundary point(s) from Step 1 & 3 on a number line.
⇒⇒ Use an open circle ALL restrictions.
⇒⇒ Use an open circle when the given inequality has << or >>
⇒⇒ Use a closed circle when the given inequality has ≤≤ or ≥≥. - Choose one number, which is called a test value, from each interval and test the intervals by evaluating the given inequality at that number.
⇒⇒ If the inequality is TRUE, then the interval is a solution of the inequality.
⇒⇒ If the inequality is FALSE, then the interval is not a solution of the inequality. - Write the solution set (usually in interval notation), selecting the interval(s) from Step 5.
Example: Solving Rational Inequality using Boundary Value Method
Solve the rational inequality using boundary value method. Graph the solution set and write the solution in interval notation.
2x−1x+2>12x−1x+2>1
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Solve the rational inequality using boundary value method. Graph the solution set and write the solution in interval notation.
5xx−3≤2
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Solve the rational inequality using boundary value method. Graph the solution set and write the solution in interval notation.
4x−5x+1≥0
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Candela Citations
CC licensed content, Original
- Solving Rational Inequalities. Authored by: Michelle Eunhee Chung. Provided by: Georgia State University. License: CC BY: Attribution