Solving Rational Inequalities (2.13.2)

Question 1

• Solving inequalities:
• Factor where possible.
• Determine the points where each factor equals zero.
• Determine the points that make the denominator equal zero and must be excluded from the domain for x.
• Determine what intervals produce results that satisfy the inequality as originally stated.

Answer 1

• Solving inequalities:
• Factor where possible.
• Determine the points where each factor equals zero.
• Determine the points that make the denominator equal zero and must be excluded from the domain for x.
• Determine what intervals produce results that satisfy the inequality as originally stated.

Question 2

• Remember to carefully watch that you distribute and use signs correctly.

Answer 2

• Remember to carefully watch that you distribute and use signs correctly.

Question 3

The mission, in this problem, is to find all values of x so that
the given expression has a value greater than or equal to 0.
Factoring the numerator helps find the places where the
expression will equal 0. In this problem, the expression
equals 0 when x equals –3 and when x equals 2.
Also note that the denominator equals 0 when x equals 4.
Therefore, exclude 4 from the domain for x because the
expression is undefined at that point.
Mark on the number line the points where the expression
equals 0 or is undefined; i.e., –3, 2, and 4 for this example.
The inequality wants values that give the expression a value
greater than 0. So, pick from each region and see:
–5 produces –14/9, 0; it works.
+3 produces –6, 0; it works.
The solution interval covers all the numbers between and
including –3 and +2 plus all the numbers larger than, but not
including, +4.
Both the number line graph and the union notation clearly
state the solution values.

Answer 3

The mission, in this problem, is to find all values of x so that
the given expression has a value greater than or equal to 0.
Factoring the numerator helps find the places where the
expression will equal 0. In this problem, the expression
equals 0 when x equals –3 and when x equals 2.
Also note that the denominator equals 0 when x equals 4.
Therefore, exclude 4 from the domain for x because the
expression is undefined at that point.
Mark on the number line the points where the expression
equals 0 or is undefined; i.e., –3, 2, and 4 for this example.
The inequality wants values that give the expression a value
greater than 0. So, pick from each region and see:
–5 produces –14/9, 0; it works.
+3 produces –6, 0; it works.
The solution interval covers all the numbers between and
including –3 and +2 plus all the numbers larger than, but not
including, +4.
Both the number line graph and the union notation clearly
state the solution values.