4.6 Day 3

4.6 Day 3

Absolute Value Inequalities

Absolute Value Inequalities (greatOR)

•  $\left|x\right|>5$  

• Distance from zero is greater than 5

• This includes numbers bigger than 5 or numbers smaller than -5

• It would be written  $x>5\ or\ x<-5$  

Another Example:

 $\left|w\right|\ge10$  

• Write two inequalities (similar to absolute value equations)

• bigger than or equal to 10 or smaller than or equal to -10

•  $w\ge10\ or\ w\le-10$  

• Change the second inequality and sign of number

Multiple Choice

Write  the following inequality without the absolute value.

 $\left|p\right|>4$  

•  $p>4$  or  $p>-4$  

•  $-4<p<4$  

•  $p>4$ or  $\ p<-4$  

•  $-4<p>4$  

Graph

 $\left|x\right|\ge3$  

graphs just like an "or" from compound inequalities
 $x\le-3\ or\ x\ge3$  

Multiple Choice

Graph

 $\left|h\right|>5$  

Absolute Value Inequality (Less ThAND)

Similar to "and" compound inequalities

Less ThAND:

 $\left|x\right|<3$  

• Distance from zero is less than 3

• This includes numbers smaller than 3 but also bigger than -3

• It would be written:  $x<3\ and\ x>-3$  

• Better written as:   $-3<x<3$  

Another example:

 $\left|x\right|\le5$  

• Becomes:   $-5\le x\le5$  

• Graphs like an AND from section 4.5

Multiple Choice

| x | ≤ 3.5

• No Solution

• All Real Numbers

• -3.5 ≤ x ≤ 3.5

• x ≤ -3.5 or x ≥ 3.5

Solving Absolute Value Inequalities

• Decide if OR or AND

• Write out Inequality

• Solve each part

Solve:

 $\left|w-5\right|\le18$  

• Less than means AND

• Write the expression  $w-5$   in between -18 and 18

• Solve (like in section 4.5)

Multiple Choice

| x + 3 | > 8

• No Solution
• All Real Numbers
• x < -11  or  x > 5
• -5 < x < 11

Multiple Choice

| x + 1 | ≥ 3

• No Solution
• All Real Numbers
• x ≤ -4  or  x ≥ 2
• -4 ≤ x ≤ 2

Multiple Choice

| 2w - 1 | < 11

• -5 < w < 6

• w < 6 or w > -5

• -6 < w < 5

Multiple Choice

| x + 3 | ≤ 2

• $-5\le x\le-1$

• $x\ge-1\$ or $x\le-5$
  

• $1\le x\le5$