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Equations and Inequalities

Linear Inequalities and Interval Notation. Section 1.5. Section 1.5. Section 1.5. Equations and Inequalities. Interval Notation. ) or ( means “not equal to” or not inclusive ] or [ means “equal to” or inclusive ±∞ always gets a parentheses

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Equations and Inequalities

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  1. Linear Inequalities and Interval Notation Section 1.5 Section 1.5 Section 1.5 Equations and Inequalities

  2. Interval Notation • ) or ( means “not equal to” or not inclusive • ] or [ means “equal to” or inclusive • ±∞ always gets a parentheses • Written with smallest desired number on left, largest desired number on the right. • Example

  3. The solutions are all real numbers less than 2. A parenthesis is used in the graph to indicate 2 is not a solution. ) EXAMPLE Graph simple inequalities Many times instead of using inequality symbols we will use a new notation called Interval Notation… Graph x < 2. Instead of using open/closed dots, we will now use parenthesis and brackets to indicate exclusive/inclusive. Just like interval notation.

  4. The solutions are all real numbers greater than or equal to –1. A bracket is used in the graph to indicate –1is a solution. [ EXAMPLE Graph simple inequalities Interval Notation: Graph x ≥ –1.

  5. The solutions are all real numbers that are greater than –1 and less than 2. ( ) EXAMPLE Graph compound inequalities Graph –1 < x < 2. Interval Notation:

  6. The solutions are all real numbers that are less than or equal to –2 or greater than 1. ] ( EXAMPLE Graph compound inequalities Graph x ≤ –2 orx > 1. Interval Notation: (-∞, -2] U (1, ∞) The U means “union”…the useful values can come from either interval. Many times we take it to mean “or”

  7. Graphing Compound Inequalities Rewrite the interval as a single interval if possible. (-∞, 5)∩(-2, ∞) The intersection symbol ∩ means “and”. This desired result has to satisfy BOTH intervals.

  8. Graph [-4,5)∩[-2,7)

  9. Graph (-7, 3)U[0, 5)

  10. Graph (-∞, -2]U[-2, ∞)

  11. Rewrite in interval notation and graph X ≤ 5

  12. Write the inequality in interval notation

  13. ANSWER (-∞, 20 ] EXAMPLE Solve an inequality with a variable on one side 20 + 1.5g ≤ 50. 20 + 1.5g ≤ 50 Write inequality. 1.5g ≤ 30 Subtract 20 from each side. g ≤ 20 Divide each side by 1.5.

  14. ANSWER The solutions are all real numbers less than 3. The graph is shown below. ) EXAMPLE Solve an inequality with a variable on both sides Solve 5x + 2 > 7x – 4. Then graph the solution. 5x + 2 > 7x – 4 Flip the inequality when multiplying or dividing both sides by a negative #. – 2x + 2 > – 4 – 2x > – 6 x < 3 (-∞, 3)

  15. Solve the inequality and express in interval notation

  16. Solve and write the answer in interval notation

  17. You rent a car for two days every weekend for a month. They charge you $50 per day, as well as $.10 per mile. Your bill has ranged everywhere from $135 to $152. What is the range of miles you have traveled?

  18. ANSWER ANSWER x < 4 (-∞, 4) x > – 7 [-7,∞) ANSWER ANSWER x ≤ 5 (-∞, 5] x <6 (-∞, 6) GUIDED PRACTICE Solve the inequality. Then graph the solution. 4x + 9 < 25 5x – 7 ≤ 6x 3 – x > x – 9 1 – 3x ≥ –14

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