Solving Inequalities

Tuesday, 27 February 2018

12:25 PM

Basics

• You can add or subtract the same term from both sides
• You can always multiply or divide both sides by a positive quantity
• You cannot multiply or divide by zero
• If multiplying or dividing by a negative quantity, the direction of the inequality sign must be changed

 

 

Often, solving an inequality turns into solving an equality
 

Example Problems

1. Solve the quadratic inequality
   
    

 

2. Solve the rational inequality

Firstly…

Now, multiply both sides by the product of the square of both denominators…

 

 

3. How small does  need to become so that
   Think of || as the .
   Reworded question: "How close does x have to be to 2 so that x2 is within 1 of 4"

 

 

 

 

 

 

 

 

 

 

 

1. Solve the inequality

Geometric Solution

The distance from -2 to x is greater than or equal to 4

• Take 4 units from the zero of  // x = -2
• Therefore x <= - 6 or x >= 2

Algebraic Solution

The inequality  is equivalent to  or
These are equivalent to and

 

2. Solve the inequality

Algebraic Solution

1. Suppose that x =/= 1
2. Multiply both sides of the inequality by the positive number |x-1| and obtain |x+5|<|x-1|
3. Square both (positive) sides ==> . Expanding and solving, x < -2

Geometric Solution

The distance from  to  is less than the distance from  to . In other words,  is closer to  than to .