Absolute Value Equations and Inequalities (A1 1-7)

Absolute Value Equations and Inequalities

(A1 1-7)

by Erica Nafziger

​Essential Question

​Why does the solution for an absolute value equation or inequality typically result in a pair of equations or inequalities?

Absolute Value ​EQUATIONS

​

Multiple Select

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

$$|x+5|=12$$  

• |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 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

• Divide both sides by 5

• Subtract 12 from both sides

• Subtract 5 from both sides

• Add 12 to both sides

Multiple Select

What are the two solutions for x?  

$$x+5=\ 12\ \ \ or\ \ \ x+5=-12$$  

• x = 7

• x = -17

• x = -7

• x = 17

Multiple Choice

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

$$4\left|x+8\right|=36$$  

• add 4 to both sides

• subtract 6 from both sides

• divide by 4 on both sides

• add 6 to both sides

Multiple Select

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

$$\frac{\left(4∣x+6∣\right)}{4}​=\frac{36}{4}$$ ----->   $$​\left|x\ +\ 6\right|=9$$  

• |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 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? 

  $$​\left|x\ +\ 6\right|=9$$  -----> $$x+6=9\ \ \ and\ \ \ x+6=-9$$  

• x = 3

• x = -15

• x = -3

• x = 15

​Absolute Value INEQUALITIES

​

Multiple Choice

Which symbol means less than?

• $$<$$  

• $$>$$  

• $$\le$$  

• $$\ge$$  

Multiple Choice

Which symbol means greater than or equal to?

• $$<$$  

• $$>$$  

• $$\le$$  

• $$\ge$$  

​Greater than (>)

​&

Greater than equal to (>)

​

*Remember GREAT "OR"

​Less than ( <)

​&

Less than equal to ( <)

​

*Remember LESS "AND"

Multiple Choice

Would this absolute value equation be an "AND" or "OR"?  $$\left|x-5\right|\le2$$  

• AND

• OR

Multiple Choice

Would this absolute value equation be an "AND" or "OR"? 

$$\left|-4x+7\right|\ge3$$  

• AND

• OR

Multiple Choice

Would this absolute value equation be an "AND" or "OR"? 

$$∣3x+1∣<7$$  

• AND

• OR

Multiple Select

Which of the following absolute value inequalities would be an "OR"?

(CHOOSE ALL THAT APPLY)

• $$\left|4x-1\right|>5$$  

• $$∣5x+\frac{1}{2}∣​\ge1$$  

• $$∣7x+5∣>6$$  

• $$\left|-3x+7\right|<11$$  

• $$∣2x−10∣\le50$$  

Multiple Select

Which of the following absolute value inequalities would be an "AND"?

(CHOOSE ALL THAT APPLY)

• $$\left|4x-1\right|>5$$  

• $$∣5x+\frac{1}{2}∣​\ge1$$  

• $$∣7x+5∣>6$$  

• $$\left|-3x+7\right|<11$$  

• $$∣2x−10∣\le50$$  

Multiple Choice

Is this absolute value inequality going to be an "AND" or an "OR"?

  $$∣x+6∣<2$$  

• AND

• OR

Multiple Select

Which two inequalities below would be the next step in solving 

  $$∣x+6∣<2$$  

• $$x+6<−2$$  

• $$x+6<2$$  

• $$x+6=2$$  

• $$x+6>-2$$  

• $$x+6>2$$  

Multiple Choice

Solve the absolute value inequality.

  $$∣2x−1∣<3$$  

• $$x<−1\ or\ x>2$$  

• $$−1<x<2$$  

• $$1<x<2$$  

• $$−1>x>2$$  

Multiple Choice

Is this absolute value inequality going to be an "AND" or an "OR"?

  $$9∣x+4∣+10\le55$$  

• AND

• OR

Open Ended

What would be the first step to isolate the absolute value?

  $$9∣x+4∣+10\le55$$  

Multiple Choice

What would be the next step to isolate the absolute value? 

  $$9∣x+4∣\le45$$  

• Divide by 9

• Subtract 4

• Add 9

• Multiply by 9

Multiple Select

Which two inequalities below would be the next step? Click two below

  $$∣x+4∣\le5$$  

• $$x+4\le−5$$  

• $$x+4\le5$$  

• $$x+4\ge5$$  

• $$x+4\ge−5$$  

Multiple Choice

Solve the absolute value inequality.

$$6\left|x+3\right|-10\ge-4$$  

• $$x\le−4\ or\ x\ge−2$$  

• $$-2<x<4$$  

• $$−4<x<−2$$  

• $$x\le−4\ or\ x\ge2$$  

Open Ended

Write in your own words the steps to solve absolute value inequalities.