Absolute Value Equations and Inequalities

Absolute Value Equations and Inequalities

Multiple Select

What is the first step in solving this absolute value equation? 

 $|x+5|=12$  

(select all that apply) 

• |x+5| = 12

  
  Set the absolute value equal to 12 

• |x+5| = -12

  
  Set the absolute value equal to -12 
  

• x+5 = 12

  
  Set the expression inside the absolute value equal to 12
  

• x+5 = -12

  
  Set the expression inside the absolute value equal to -12
  

Multiple Choice

After we set the expression equal to 12 and -12, what is the second step? 

 $|x+5|=12$  
 $x+5=12\$  
or 
 $x+5\ =-12$  

• add 5 to both sides 

• subtract 5 from both sides 

• divide both sides by 5   

• add 12 to both sides

• subtract 12 from both sides

Multiple Select

What are the two solutions for x?  

 $x+5=12\$  
or
 $\ x+5\ =-12\$  

(select all that apply) 

• x = 7

• x = -7 

• x = -17   

• x = 17

Multiple Choice

What is the first step in solving this absolute value equation? 

 $4|x+8|=36\ \$  

*Hint: we need to isolate the absolute value. 

• add 4 to both sides 

• divide by 4 on both sides

• subtract 6 from both sides

• add 6 to both sides

Multiple Select

After we isolate the absolute value, what do we do next?

 $\frac{4|x+6|}{4}=\frac{36}{4}\$  
 $\left|x\ +\ 6\right|=9$  

(select all that apply)

• |x+6|=9

  
  set the absolute value equal to 9

• |x+6|=-9

  
  set the absolute value equal to -9
  

• x+6=9

  
  set the  expression inside the absolute value equal to 9
  

• x+6=-9

  
  set the  expression inside the absolute value equal to -9
  

Multiple Select

What are the two solutions for x? 

 $\left|x\ +\ 6\right|=9$   
 $x+6=9\$  
or
 $x+6=-9$  
(select all that apply) 

• x = 3

• x = -3 

• x = -15

• x = 15

Multiple Choice

What is the first step in solving this absolute value inequality? $\left|2x+5\right|-6\le11\$  

• divide by 2 on both sides

• subtract 5 from both sides

• add 6 to both sides

• set the equation between -11 and 11

Multiple Choice

Once we isolate the absolute value, how to we set up the compound inequality? 
 $\left|2x+5\right|-6\le11\$  
 $\left|2x+5\right|\le17$  

•  $-17\le2x+5\le17$  
  

•  $2x+5\le17$  

•  $2x+5\le-17$  

•  $-17\le\left|2x+5\right|\le17$  

Multiple Choice

Solve for x. 

 $-17\le2x+5\le17$  

•  $-6\le x\le6$  
  

•  $-22\le x\ \le12$  
  

•  $11\ge x\ \ge-6$  

•   $-11\le x\ \le6$  

Multiple Choice

What is the first step in solving this absolute value inequality? $3\left|x-4\right|+2<23$  

• subtract 2 from both sides

• divide by 3 on both sides

• add 4 to both sides

• set the equation between -23 and 23

Multiple Choice

What do we do next?

 $3\left|x-4\right|+2<23$  

 $3\left|x-4\right|<21$  

• set the equation between -21 and 21

• divide by 3 on both sides

• add 4 to both sides

Multiple Choice

What do we do next?

 $3\left|x-4\right|+2<23$  

 $3\left|x-4\right|<21$  

 $\left|x-4\right|<7$  

•  $-7<\left|x-4\right|<7\$  

•  $-7<x-4<7\$  
  

•  $x-4<7\$  
  

•  $x-4>-7$  

Multiple Choice

Solve for x. 

 $3\left|x-4\right|+2<23$  

 $3\left|x-4\right|<21$  

 $\left|x-4\right|<7$  
 $-7<x-4<7\$  

•  $3<x<11$  

•  $-11<x<3$  
  

•  $-3<x<11$  

•  $-11<x<11$  

Multiple Select

How do we set up this absolute value inequality? 
 $\left|3x+6\right|\ge18$  

(select all that apply) 

•  $3x+6\ge18$  
  

  set the expression to greater than or equal to 18

•  $-18\le3x+6\le18$  
  

  set the expression value between -18 and 18 

•  $3x+6\le-18$  

  set the expression to less than or equal to -18
  

•  $3x+6\ge-18$  
  set the expression to greater than or equal to -18

Multiple Select

Solve for x.
 $\left|3x+6\right|\ge18$  

 $3x+6\le-18$ 
or
 $3x+6\ge18\$  
 (select all that apply) 

•  $x\ge8$  
  

•  $x\le-8$  
  

•  $x\ge-8$  

•  $x\ge4$   
  

•  $x\le4$  

Multiple Choice

What are the first two steps?
 $4\left|x-4\right|+7>31$  

• First, subtract 7, then multiply by 4 on both sides

• First, divide by 4, then subtract 7 from both sides

• First, subtract 7, then divide by 4 on both sides
  

Multiple Choice

 How do we set up the inequality after we isolate the  absolute value?
 $4\left|x-4\right|+7>31$  
 $4\left|x-4\right|>24$ 

 $\left|x-4\right|>6$  

•  $x-4>-6\$  

  
   $x-4>6$  
  

•  $x-4<-6\$  
  
   $x-4>6$  
  

•   $-6<x-4<6$  

Multiple Select

Solve for x. 

 $\left|x-4\right|>6$  
 $x-4<-6\$  or
 $x-4>6$  
 (select all that apply) 

•  $x\ <\ -10\$  

•  $x<-2$  

•  $x\ >\ 2$  

•  $x>10$