Solving Quadratic Inequalities: Another Example (2.12.2)

Question 1

• Solving inequalities:
• Factor where possible.
• Determine the points where each factor equals zero.
• 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 what intervals produce results that satisfy the inequality as originally stated.

Question 2

• Remember that multiplying or dividing by a negative number reverses the inequality.

Answer 2

• Remember that multiplying or dividing by a negative number reverses the inequality.

Question 3

• Remember that graphing on a number line is the best way to spot all the possibilities.

Answer 3

• Remember that graphing on a number line is the best way to spot all the possibilities.

Question 4

Your original inequality asks for all the values of x that make
3x
2
smaller than (–10 – 13x).
To solve, do your usual steps for solving a quadratic equation:
1. Move everything to one side and set it equal to 0.
2. Factor if possible.
3. Set each factor equal to 0 and solve for x.
Then do your steps for inequalities:
1. Set up a number line with the points marked
where the factors equal 0. In this case, that
means you mark –10/3 and –1. Setting two
points creates three intervals on the number line.
2. Choose a random point from each interval
on the number line.
3. Substitute that point for x in the original equation.
4. Check to see which intervals yield the results
indicated by the original equation.
For example, choose and test with:
–5: (–15 + 10)(–5 + 1) = 20, > 0.
–2: (–6 + 10)(–2 + 1) = –4, 0.
Mark the number line and/or use other notation to show which
intervals are solutions.

Answer 4

Your original inequality asks for all the values of x that make
3x
2
smaller than (–10 – 13x).
To solve, do your usual steps for solving a quadratic equation:
1. Move everything to one side and set it equal to 0.
2. Factor if possible.
3. Set each factor equal to 0 and solve for x.
Then do your steps for inequalities:
1. Set up a number line with the points marked
where the factors equal 0. In this case, that
means you mark –10/3 and –1. Setting two
points creates three intervals on the number line.
2. Choose a random point from each interval
on the number line.
3. Substitute that point for x in the original equation.
4. Check to see which intervals yield the results
indicated by the original equation.
For example, choose and test with:
–5: (–15 + 10)(–5 + 1) = 20, > 0.
–2: (–6 + 10)(–2 + 1) = –4, 0.
Mark the number line and/or use other notation to show which
intervals are solutions.