Here are the steps required for Solving Rational Inequalities:

Step 1: Write the inequality in the correct form. One side must be zero and the other side can have only one fraction, so simplify the fractions if there is more than one fraction.
Step 2: Find the key or critical values. To find the key/critical values, set the numerator and denominator of the fraction equal to zero and solve.
Step 3: Make a sign analysis chart. To make a sign analysis chart, use the key/critical values found in Step 2 to divide the number line into sections.
Step 4: Perform the sign analysis. To do the sign analysis, pick one number from each of the sections created in Step 3 and plug that number into the polynomial to determine the sign of the resulting answer. The sign of this answer (positive or negative) will be sign of the entire section. You can check different number from the same section if you want to verify your answer.
Step 5: Use the sign analysis chart to determine which sections satisfy the inequality. If the inequality is less than zero or less than or equal to zero, then you want all of the negative sections found in the sign analysis chart. If the inequality is greater than zero or greater than or equal to zero, then you want all of the positive sections found in the sign analysis chart.
Step 6: Use interval notation to write the final answer.

Example 1 – Graph: Example 1

Step 1: Write the inequality in the correct form. One side must be zero and the other side can have only one fraction, so simplify the fractions if there is more than one fraction.
Step 1
Step 2: Find the key or critical values. To find the key/critical values, set the numerator and denominator of the fraction equal to zero and solve.
Step 2
Step 3: Make a sign analysis chart. To make a sign analysis chart, use the key/critical values found in Step 2 to divide the number line into sections.
Step 3
Step 4: Perform the sign analysis. To do the sign analysis, pick one number from each of the sections created in Step 3 and plug that number into the polynomial to determine the sign of the resulting answer. In this case, you can choose x = –5 which results in –1.333, x = 0 which results in +12, x = 2 which results in –6, and x = 4 which results in +2.666.
Step 4
Step 5: Use the sign analysis chart to determine which sections satisfy the inequality. In this case, we have less than or equal to zero, so we want all of the negative sections. Notice that x ≠ 1 because it would make the original problem undefined, so you should use an open circle at x = 1 instead of a closed circle to draw the graph.
Step 5
Step 6: Use interval notation to write the final answer.
Step 6

Example 2 – Graph: Example 2

Step 1: Write the inequality in the correct form. One side must be zero and the other side can have only one fraction, so simplify the fractions if there is more than one fraction.
Step 1
Step 2: Find the key or critical values. To find the key/critical values, set the numerator and denominator of the fraction equal to zero and solve.
Step 2
Step 3: Make a sign analysis chart. To make a sign analysis chart, use the key/critical values found in Step 2 to divide the number line into sections.
Step 3
Step 4: Perform the sign analysis. To do the sign analysis, pick one number from each of the sections created in Step 3 and plug that number into the polynomial to determine the sign of the resulting answer. In this case, you can choose x = –4 which results in –0.025, x = 0 which results in +0.75, x = 2 which results in –2.5, and x = 4 which results in +2.
Step 4
Step 5: Use the sign analysis chart to determine which sections satisfy the inequality. In this case, we have greater than or equal to zero, so we want all of the positive sections. Notice that x ≠ 1 and x ≠ 4 because it would make the original problem undefined, so you should use an open circle at x = 1 and x = 4 instead of a closed circle to draw the graph.
Step 5
Step 6: Use interval notation to write the final answer.
Step 6

Click Here for Practice Problems

Example 3 – Graph:Graph: Example 3

Step 1: Write the inequality in the correct form. One side must be zero and the other side can have only one fraction, so simplify the fractions if there is more than one fraction.
Step 1
Step 2: Find the key or critical values. To find the key/critical values, set the numerator and denominator of the fraction equal to zero and solve.
Step 2
Step 3: Make a sign analysis chart. To make a sign analysis chart, use the key/critical values found in Step 2 to divide the number line into sections.
Step 3
Step 4: Perform the sign analysis. To do the sign analysis, pick one number from each of the sections created in Step 3 and plug that number into the polynomial to determine the sign of the resulting answer. In this case, you can choose x = –6 which results in –0.031, x = –3 which results in +0.4, x = 0 which results in –1.25, and x = 3 which results in +1.6.
Step 4
Step 5: Use the sign analysis chart to determine which sections satisfy the inequality. In this case, we have less than zero, so we want all of the negative sections.
Step 5
Step 6: Use interval notation to write the final answer.
Step 6

Click Here for Practice Problems

Example 4 – Graph:Graph: Example 4

Step 1: Write the inequality in the correct form. One side must be zero and the other side can have only one fraction, so simplify the fractions if there is more than one fraction.
Step 1
Step 2: Find the key or critical values. To find the key/critical values, set the numerator and denominator of the fraction equal to zero and solve.
Step 2
Step 3: Make a sign analysis chart. To make a sign analysis chart, use the key/critical values found in Step 2 to divide the number line into sections.
Step 3
Step 4: Perform the sign analysis. To do the sign analysis, pick one number from each of the sections created in Step 3 and plug that number into the polynomial to determine the sign of the resulting answer. In this case, you can choose x = –2 which results in +0.25, x = 0 which results in –0.5, and x = 3 which results in +4.
Step 4
Step 5: Use the sign analysis chart to determine which sections satisfy the inequality. In this case, we have less than or equal to zero, so we want the negative section. Notice that x ≠ 2 because it would make the original problem undefined, so you must use an open circle at x = 2 instead of a closed circle to draw the graph.
Step 5
Step 6: Use interval notation to write the final answer.
Step 6

Click Here for Practice Problems

Example 5 – Graph:Graph:

Step 1: Write the inequality in the correct form. One side must be zero and the other side can have only one fraction, so simplify the fractions if there is more than one fraction.
Step 1
Step 2: Find the key or critical values. To find the key/critical values, set the numerator and denominator of the fraction equal to zero and solve.
Step 2
Step 3: Make a sign analysis chart. To make a sign analysis chart, use the key/critical values found in Step 2 to divide the number line into sections.
Step 3
Step 4: Perform the sign analysis. To do the sign analysis, pick one number from each of the sections created in Step 3 and plug that number into the polynomial to determine the sign of the resulting answer. In this case, you can choose x = –4 which results in +0.2, x = 0 which results in –3, and x = 2 which results in +5.
Step 4
Step 5: Use the sign analysis chart to determine which sections satisfy the inequality. In this case, we have greater than or equal to zero, so we want all of the positive sections. Notice that x ≠ 1 because it would make the original problem undefined, so you must use an open circle at x = 1 instead of a closed circle to draw the graph.
Step 5
Step 6: Use interval notation to write the final answer.
Step 6

Click Here for Practice Problems