Linear inequalities and absolute value

Section 3-1: Linear inequalities and Absolute Value

Solving a linear inequality is very similar to solving a linear equation. You can basically solve it the same way except for one situation. When you multiply/divide both sides of an inequality by a negative number, change the sense (direction) of the inequality. Sample problems

Problem 1
   3x - 5 < 9 + x
   2x - 5 < 9
        2x < 14
          x < 7

Problem 2
       8 - 2x > 6
           -2x > -2
              x < 1

In problem 1, it was solved exactly like an equation. The solution is all numbers less than 7 make the statement true. The graph looks like: The open dot means does not include!

In problem two, notice we changed the sense in the last step because we divided both sides by -2. The graph looks like: The closed dot means it includes this point!

Absolute Value Recall that absolute value means the distance from any point to zero. Study the following chart:

Sentence	Meaning	Graph	Solution
\|x\| = a	The distance from x to 0 is exactly a units		x = a or x = -a
\|x\| < a	The distance from x to 0 is less than a units.		-a < x < a
\|x\| > a	The distance from x to 0 is greater than a units		x < -a or x > a

Sample Problems 2|3x - 5| = 8 This is an equality, choice number 1 on the chart. First, islolate the absolute value to use the chart above. Divide both sides by 2. |3x - 5| = 4 Now apply the definition in the last column. 3x - 5 = 4 or 3x - 5 = -4 Solve each one individually. Add 5 and divide by 3. 3x = 9 or 3x = 1 x = 3 or x = 1/3

3|2x + 3| < 12 This is an inequality involving the less than symbol. Use choice two from the chart. Remember to isolate the absolute value first! Divide by 3 |2x + 3| < 4 Now apply rule #2 from the above chart. -4 < 2x + 3 < 4 Isolate for x in the middle by subtracting 3 and dividing by 2 -7 < 2x < 1 -7/2 < x < 1/2 The solution is all numbers between -7/2 and 1/2.

|x - 4|/3 > 2 This is the last choice involving a greater than symbol. Again, isolate the absolute value before you apply the appropriate rule! Multiply by 3 |x - 4| > 6 Now apply the last rule from the chart. x - 4 > 6 or x - 4 < -6 Add 4 to both sides. x > 10 or x < -2 All numbers greater than or equal to 10 work, and all numbers less than or equal to -2 work!

A Slightly tougher problem!! 1 < |x| < 5 Method 1. Split into two inequalities 1 < |x| and |x| < 5 Solve individually |x| > 1 and |x| < 5 The first inequality is rule 3 and the second is rule 2. Apply these rules: x > 1 or x < -1 and -5 < x < 5 To find the solution, find the area which is common to both.

Look at these to graphs and what points do they have in common? That's right, the numbers between -1 and -5 and the numbers between 1 and 5!! The solution looks like:

Let's head on to the next section: