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# Absolute Inequalities

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• 1. Practice Problem
4a – 7 ≤ 17 AND 14 – a > -5a + 3
Solve the Compound Inequality AND Graph
• 2. Practice Problem
Rachel is planning a wedding for 100 to 250 people, depending on costs. She has hired a caterer that charges \$20 per person, plus a flat fee of \$200.
Write an inequality in terms of “p” that describes how many people Rachel plans to invite.
Adjust the inequality to show how much the caterer will charge.
What is the range of catering fees that Rachel is considering.
• 3. Solving Absolute Value Equations & Inequalities
• 4. Absolute Value (of x)
Symbol lxl
The distance x is from 0 on the number line.
Always positive
Ex: l-3l=3
-4 -3 -2 -1 0 1 2
• 5. Ex: x = 5
What are the possible values of x?
x = 5 or x = -5
• 6. To solve an absolute value equation:
ax+b = c, where c>0
To solve, set up 2 new equations, then solve each equation.
ax+b = c or ax+b = -c
** make sure the absolute value is by itself before you split to solve.
• 7. Ex: Solve 6x-3 = 15
6x-3 = 15 or 6x-3 = -15
6x = 18 or 6x = -12
x = 3 or x = -2
• 8. Ex: Solve 2x + 7 -3 = 8
Get the abs. value part by itself first!
2x+7 = 11
Now split into 2 parts.
2x+7 = 11 or 2x+7 = -11
2x = 4 or 2x = -18
x = 2 or x = -9
Check the solutions.
• 9. Solving Absolute Value Inequalities
ax+b < c, where c > 0
Becomes an “and” problem
Changes to: –c < ax+b < c
ax+b > c, where c > 0
Becomes an “or” problem
Changes to: ax+b > c or ax+b < -c
• 10. SOLVING ABSOLUTE-VALUE EQUATIONS AND INEQUALITIES
means
means
means
means
means
means
means
means
means
means
ax b  c andax b   c.
|ax b |  c
|ax b |  c
ax b  c andax b   c.
When an absolute value is less than a number, the
inequalities are connected by and. When an absolute
ax b  c orax b   c.
|ax b |  c
value is greater than a number, the inequalities are
connected by or.
ax b  c orax b   c.
|ax b |  c
ax b  c orax b   c.
|ax b |  c
• 11. Solving an Absolute Value Inequality
• Step 1: Rewrite the inequality as a conjunction or a disjunction.
• 12. If you have a < or ≤ you are working with a conjunction or an ‘and’ statement.
Remember: “Less thand”
• If you have a > or ≥you are working with a disjunction oran ‘or’ statement.
Remember: “Greator”
• Step 2: In the second equation you must negate the right hand side and reversethe direction of the inequality sign.
• 13. Solve as a compound inequality.
• Ex: Solve & graph.
Becomes an “and” problem
-3 7 8
• 14. Solve & graph.
Get absolute value by itself first.
Becomes an “or” problem
-2 3 4