Absolute Value Inequalities

Absolute Value Inequalities

How to solve when there are many solutions

Absolute Value Inequalities?

How are they different than Abs. Value Equations?

Absolute Value Inequalities:

• Use $<,\le,>,\ge$ symbols 

• Have many solutions--not just 1 or 2

• Are solved by creating 2 cases just like AV equations

• We must graph answers in some cases 

Pay attention to the Sign...

• If the original inequality is  $>$  or  $\ge$ use OR in the answer 

•         OR means we need to include all values we graph

• If the inequality is $<$  or $\le$  use AND in the answer

•       AND means we only include the INTERSECTION of the graphs

Multiple Choice

Which word should we use when the sign is:    $>,\ge$  (Greater than, greater than or equal to)?

• AND

• OR

Multiple Choice

Which word should we use when the sign is:    $<,\le$  (Less than, less than or equal to)?

• AND

• OR

Here are the steps to solving Absolute Value Inequalities:

• Step 1: Isolate the absolute value. Get the | | on one side.

• Step 2: Set up 2 inequalities. Case 1: write just like you see without the bars. The 2nd case, "flip the sign and change the sign" on the right side.

• Step 3: Solve each inequality.

• Step 4: Graph each and write the solution using AND or OR.

Multiple Choice

How many cases (inequalities) will we have to write out and solve? (Hint: how many did we have for Abs. Value Equations?)

• 2

• 0

• 1

• 5

Multiple Choice

Let's solve:      $2\left|x-5\right|>10$  First do Step 1:  Isolate the | |.

What will we have to do?

• Subtract 2

• Divide by 2

• Add 4

• Ready to make 2 cases!

Step2:  Write 2 inequalities.  

• First we need to decide whether we will use AND or OR.

• The sign was  $>$  ......

Multiple Choice

For this problem, $\left|x-5\right|>5$  , the sign is  $>$  .  Which word should we use?

• AND

• OR

Here are the 2 cases we write out:

•  $x-5>5$         OR      **** $x-5<-5$  

• ***2nd case:  The right-hand side changes  to  $<-5$  

• IMPORTANT!  For the 2nd case, always 'flip the inequality and change the sign to  '-' negative!

Multiple Choice

                      Step 3:   We'll solve each of them.

               What is the solution to the first:      $x-5>5$  

• x > 25

• x > 0

• x > 10

Multiple Choice

                           Now solve the second one.

               What is the solution to:   $x-5<-5$  

• x < 16

• x < 0

• x < -8

Step 4: Graph & write the solutions.

• <===============(0)......................................(10)==================>

• Our solutions:         $x<0\ \ \ OR\ \ \ x>10$  

  
  

Let's try another!

Solve:

 $5\left|x+2\right|-1<14$  

Multiple Choice

First, isolate:      $5\left|x+2\right|-1<14$   The new inequality will be:
                     (Hint:  'move' two things!) 

•  $\left|x+2\right|<10$  

•  $\left|x+2\right|<\frac{13}{5}$  

•  $\left|x+2\right|<9$  

•  $\left|x+2\right|<3$  

Multiple Choice

What will be the 2 inequality cases for:         $\left|x+2\right|<3$  ?  

(Do we use AND or OR?)

•  $x+2<3\ \$  AND   $x+2<-3$   

•  $x+2<3$   AND   $x+2>-3$  

•  $x+2<3$  OR   $x+2>-3$  

Remember: Use AND here and 'flip the sign and change the sign' for the 2nd case!!

Multiple Choice

Okay, now solve each inequality:     $x+2<3$  AND   $x+2>-3$  

• x < 1  AND  x > -5

• x < 5 AND x > -5

Now graph both of the solutions on a number line:

• Notice the graphs will INTERSECT from -5 to 1.

• <.....................(-5)===============(1)...............................>

• We use AND here because we want values that work in both the first case AND the second case.

• NOTE: -5 and 1 are NOT included as part of the answer because the problem was < only. There are OPEN DOTS on these 2 values.

Multiple Choice

How do we write the final solution?  (everything between -5 and 1)? 

•  $-5<x<1$  

•  $-5>x>1$  

Now's it your turn!

Here are the steps again:

• Step 1: Isolate the absolute value. Get the | | on one side.

• Step 2: Set up 2 inequalities. One just like you see without the bars. The other, "flip the sign and change the sign"

• Step 3: Solve each inequality.

• Step 4: Graph each and write the solution using AND or OR.

Multiple Choice

        Solve:  $\left|x+7\right|>13$  

• x > 6  OR  x < -20

• x < 6  OR  x < -20

• x > 20 OR x < -6

• No solution

Multiple Choice

        Solve:  $4\left|2x-1\right|\ge28$  

•  $x\ge4$  OR  $x\le-3$  
  

•  $x\le4$  OR   $x\ge-3$  

•  $x\ge4$  AND   $x\le-3$ 

•  $-3\le x\le4$  

Multiple Choice

        Solve:  $\left|3x-6\right|+4\le16$  

•  $x\ge6$  OR  $x\le-2$  
  

•    $-6\le x\le2$  

•  $x\ge-2$  OR   $x\ge6$ 

•  $-2\le x\le6$  

Multiple Choice

Solve:     $\left|x+4\right|\le-6$  **Think!

• No solution

• All Reals

Multiple Choice

This was unusual:    $\left|x+4\right|\le-6$ 

But think about it....the left side will always be positive.  

How many positive numbers will be less than or equal to -6 ??

• None, no solution!

• All positive values will be  $\le-6$ 
  Solution is All Reals 

Multiple Choice

Now try this one:     $\left|3x-6\right|\ge-5$  **Think!

• No solution

• All Reals

Multiple Choice

For this one:   $\left|3x-6\right|\ge-5$ 

Again, the left side will always be positive.  

How many positive numbers will be greater than or equal to -5 ??

• None, no solution!

• All positive values will be  $\ge-5$  

  
  

  Solution is All Reals!

Great Job!

No go practice some more!