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Any questions on Section 2.3 ? (Turn in your worksheet now.)

Any questions on Section 2.3 ? (Turn in your worksheet now.). Now please CLOSE YOUR LAPTOPS and turn off and put away your cell phones. Sample Problems Page Link (Dr. Bruce Johnston ). Section 2.4: Linear Inequalities.

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Any questions on Section 2.3 ? (Turn in your worksheet now.)

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  1. Any questions on Section 2.3 ?(Turn in your worksheet now.)

  2. Now please CLOSE YOUR LAPTOPS and turn off and put away your cell phones. Sample Problems Page Link (Dr. Bruce Johnston)

  3. Section 2.4: Linear Inequalities A linear inequality in one variable is an inequality that can be written in the form ax + b<c. Note: • a, b, and c are real numbers, a 0. • < symbol could be replaced by > or  or  Examples: 3x – 5 < 4 2x +3 ≥ x + 1.

  4. Interval Notation • (-5,-2) represents all the numbers in between -2 and -5 excluding -2 and -5. Inequality: -5 < x < -2 • (-5,-2] represents all the numbers in between -2 and -5 including -2 and excluding -5. Inequality: -5 < x ≤ -2 • [-5,-2) represents all the numbers in between -2 and -5 excluding -2 and including -5. Inequality: -5 ≤ x < -2 • [-5,-2] represents all the numbers in between -2 and -5 including -2 and -5. Inequality: -5 ≤ x ≤ -2

  5. Represents the set (-, 7] Or the inequality {xx  7} • Graphing solutions to linear inequalities in one variable and using the graph to figure out how to write the solution in interval notation: • Use a number line • Use a bracket (or a filled-in circle) at the endpoint of an interval if you want to include the point • Use a parenthesis (or an open circle) at the endpoint if you DO NOT want to include the point -∞ ∞ Represents the set (-4, ) Or the inequality{xx > -4} -∞ ∞

  6. Example from today’s homework:

  7. Example from today’s homework:

  8. Example from today’s homework:

  9. Addition property of inequality • a< b and a + c < b + c are equivalent inequalities. Example: 2 ≤ 4 and 2 + (-3) ≤ 4 + (-3) are equivalent Multiplication property of inequality • if c ispositive, then: a< b and ac < bc are equivalent inequalities, Example: 3 ≥ 1 (multiply both sides by 2); so 6 ≥ 2 is equivalent. • if c isnegative, then: a< b and ac > bc are equivalent inequalities, Example: 3 ≥ 1 (multiply both sides by -2); so -6 ≤ -2 is equivalent. .

  10. Solving linear inequalities in one variable • Multiply to clear fractions. • Use the distributive property (parentheses). • Simplify each side of the inequality. • Get all variable terms on one side and numbers on the other side of inequality (addition property of inequality). • Isolate variable (multiplication property of inequality). • Do not forget to change the direction of the inequality sign if you multiply or divide both sides by a negative number.

  11. Caution: Don’t forget that if both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality sign MUST BE REVERSED.

  12. Example 1: (divide both sides by -4 and simplify) Graph of solution ( ,) -7(x – 2) - x < 4(5 – x) + 12 -7x + 14 - x < 20 - 4x + 12(use distributive property) - 8x + 14 < - 4x + 32(simplify both sides) - 8x + 4x + 14 < - 4x + 4x + 32(add 4x to both sides) - 4x + 14 < 32(simplify both sides) - 4x + 14 - 14 < 32 - 14(subtract 14 from both sides) - 4x < 18(simplify both sides)

  13. Example 2:

  14. Example from today’s homework:

  15. Something to think about: • How would you graph the inequality 2 > x? • What would this look like in interval notation? Note that 2 > x is equivalent to x < 2. Writing the inequality with the variable term on the left makes it easier to “see” what the graph and the interval notation should look like. Interval notation: (-∞, 2) This is an argument for working to put/keep your variables on the left side of the expression as you solve linear inequalities.

  16. REMINDER: In interval notation, ∞ and -∞ ALWAYS are enclosed by a (round bracket) NEVER by a [ square bracket].

  17. Reminder: This homework assignment on section 2.4 is due at the start of next class period. (There’s no paper worksheet for this assignment, but you should still work the problems in your notebook as you do the online assignment.)

  18. Next Class: In the next class session we will be reviewing for Quiz 1, which will have questions from all the homework assignments we have covered through today. There is a practice quiz available that you can start before the next class if you want to get a head start on preparing for the quiz.

  19. You may now OPEN your LAPTOPS and begin working on the homework assignment.

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