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# Absolute Inequalities

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### Absolute Inequalities

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