Solving absolute value inequalities

Lessons

• Introduction
  $\bullet$ definition of absolute value: $| \heartsuit | =$ distance of $\heartsuit$ from zero
  $\bullet$ absolute value inequalities: $| \heartsuit |$ < $a$,

  solution:$-a$ < $\heartsuit$ < $a$
  $\bullet$ absolute value inequalities: $| \heartsuit |$ > $a$,

  solution:$\heartsuit$ < $-a$ $\;or\;$ $\heartsuit$ > $a$
  

---

• 1.
  Solving Basic Absolute Value Inequalities
  Solve:
  a)

  $|x|$ < $4$
  $|x| \leq$ $4$
  
  
  b)

  $|x|$ > $4$
  $|x| \geq$ $4$
  
  

---

• 2.
  Solving Absolute Value Inequalities Involving "less than"
  Solve: $|2x-1|$ < $3$

---

• 3.
  Solving Absolute Value Inequalities Involving "greater than"
  Solve:
  a)

  $|4x-5|$ > $7$
  
  
  b)

  $|x|-5 \geq -1$
  
  

---

• 4.
  Multiplying/Dividing an Inequality by a Negative Number
  Solve:
  a)

  $|3-2x| \leq 11$
  
  
  b)

  $|-\frac{x}{6}+\frac{5}{3}|$ > $2$
  
  

---

• 5.
  Given a Pair of Inequalities, Determine the Corresponding Absolute Value Inequality
  Determine the absolute value inequality statement that corresponds to each inequality:
  a)

  $-1$ < $x$ < $5$
  
  
  b)

  $x \leq-10$ $\;or\;$ $x \geq 2$
  
  

---

• 6.
  Recognizing Absolute Value Inequalities with "No Solution" or "All Real Numbers"
  Solve:
  a)

  $|x+3|$ < $-5$
  
  
  b)

  $|x-4|$ > $-1$
  
  

---