Interval Notation Rational Inequality Calculator

What is a Rational Inequality?

A rational inequality is an inequality that contains a rational expression, which is a fraction where both the numerator and denominator are polynomials. The general form is:

Where \(P(x)\) and \(Q(x)\) are polynomials. Solving these inequalities requires finding the values of \(x\) that satisfy the inequality while considering where the expression is undefined (denominator equals zero).

The solution to a rational inequality is typically expressed in interval notation, which provides a clear representation of the range of values that satisfy the inequality.

Solving Rational Inequalities

The process for solving rational inequalities involves several key steps:

1. Identify critical points: Find the values of \(x\) that make the numerator or denominator zero.
2. Determine undefined points: Identify values of \(x\) that make the denominator zero (these points are excluded from the solution).
3. Create a sign chart: Divide the number line into intervals using the critical points and test the sign of the rational expression in each interval.
4. Determine solution intervals: Based on the inequality sign, select the intervals where the expression is positive or negative.
5. Express in interval notation: Write the solution using interval notation, excluding any undefined points.

It's important to consider the inequality sign when determining which intervals to include in the solution. For example, if the inequality is \(\leq 0\), you would include intervals where the expression is negative or zero.

Interval Notation

Interval notation is a concise way to represent sets of real numbers. The common symbols used are:

• (a, b): All numbers between \(a\) and \(b\), not including \(a\) and \(b\)
• [a, b]: All numbers between \(a\) and \(b\), including \(a\) and \(b\)
• (a, b]: All numbers between \(a\) and \(b\), not including \(a\) but including \(b\)
• [a, b): All numbers between \(a\) and \(b\), including \(a\) but not including \(b\)
• (a, ∞): All numbers greater than \(a\)
• (-∞, b): All numbers less than \(b\)
• (-∞, ∞): All real numbers

When solving rational inequalities, the solution is typically expressed as a union of intervals in interval notation, excluding any points where the expression is undefined.

Worked Examples

Example 1: Solving \(\frac{x+2}{x-3} > 0\)

1. Find critical points: \(x+2=0\) → \(x=-2\), \(x-3=0\) → \(x=3\) (undefined point)
2. Create intervals: \((-∞, -2)\), \((-2, 3)\), \((3, ∞)\)
3. Test each interval:
   • For \(x=-3\): \(\frac{-3+2}{-3-3} = \frac{-1}{-6} = \frac{1}{6} > 0\) → Positive
   • For \(x=0\): \(\frac{0+2}{0-3} = \frac{2}{-3} = -\frac{2}{3} < 0\) → Negative
   • For \(x=4\): \(\frac{4+2}{4-3} = \frac{6}{1} = 6 > 0\) → Positive
4. Solution: The expression is positive on \((-∞, -2)\) and \((3, ∞)\).

Final solution in interval notation: \((-∞, -2) \cup (3, ∞)\)

Example 2: Solving \(\frac{x-1}{x+4} \leq 0\)

1. Find critical points: \(x-1=0\) → \(x=1\), \(x+4=0\) → \(x=-4\) (undefined point)
2. Create intervals: \((-∞, -4)\), \((-4, 1)\), \((1, ∞)\)
3. Test each interval:
   • For \(x=-5\): \(\frac{-5-1}{-5+4} = \frac{-6}{-1} = 6 > 0\) → Positive
   • For \(x=0\): \(\frac{0-1}{0+4} = \frac{-1}{4} = -0.25 < 0\) → Negative
   • For \(x=2\): \(\frac{2-1}{2+4} = \frac{1}{6} > 0\) → Positive
4. Solution: The expression is negative on \((-4, 1)\) and zero at \(x=1\).

Final solution in interval notation: \((-4, 1]\)