7.6 Solve Rational Inequalities

Rational inequalities involve fractions with polynomials in the numerator and denominator. Solving them requires finding critical points, identifying undefined points, and determining the sign of the function in different intervals.

Graphing solutions on a number line helps visualize the results. Open circles represent strict inequalities, closed circles show inclusive inequalities, and shading indicates where the inequality is satisfied. These skills are crucial for analyzing rational functions.

Solving Rational Inequalities

Solving rational inequalities

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• Rational inequalities contain fractions with polynomials in the numerator and denominator (e.g., $\frac{x+2}{x-1} [>](https://www.fiveableKeyTerm:>) 3$)
• Solve rational inequalities using these steps:
  1. Find critical points by setting numerator and denominator equal to zero and solving for x
  2. Identify undefined points where the denominator equals zero, as the rational function is undefined at these x-values (asymptotes)
  3. Determine the sign (positive or negative) of the rational function in each interval created by the critical points and undefined points
     • Test a point from each interval in the original inequality to determine the sign (e.g., if $x [<](https://www.fiveableKeyTerm:<) 2$, test $x = 0$)
  4. Combine the intervals where the inequality is satisfied to find the solution set (e.g., $x < -2$ or $x > 1$)

Graphing inequality solutions

• Represent the solution set of a rational inequality on a number line
  • Use an open circle (○) for strict inequalities ($<$ or $>$)
  • Use a closed circle (●) for inclusive inequalities ($\leq$ or $\geq$)
  • Shade the number line to indicate the intervals where the inequality is satisfied
    • Shade to the right for $x > a$ or $x \geq a$ (e.g., $x > 3$ shades from 3 to positive infinity)
    • Shade to the left for $x < a$ or $x \leq a$ (e.g., $x \leq -1$ shades from negative infinity to -1)
  • If the solution set consists of multiple intervals, shade each interval separately (e.g., $x < -2$ or $x > 1$ shades from negative infinity to -2 and from 1 to positive infinity)

Comparing rational functions to values

• Determine when a rational function is greater than, less than, or equal to a given value $k$:
  1. Set up an inequality comparing the rational function to $k$
     • If $f(x)$ is the rational function, set up $f(x) > k$, $f(x) < k$, or $f(x) = k$ (e.g., if $f(x) = \frac{x+1}{x-2}$, set up $\frac{x+1}{x-2} > 5$)
  2. Solve the resulting rational inequality using the method described earlier
     • Find critical points and undefined points
     • Determine the sign of the function in each interval
     • Identify the intervals where the inequality is satisfied (e.g., $x < -6$ or $x > 2$)
  3. The solution set represents the x-values for which the rational function satisfies the given condition (greater than, less than, or equal to $k$)

Properties of Rational Functions

• Domain: The set of all possible input values (x-values) for which the rational function is defined
• Range: The set of all possible output values (y-values) that the rational function can produce
• Continuity: A rational function is continuous at all points in its domain, except at vertical asymptotes