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Core Focus on Rational Numbers & Equations

Learn to solve equations and inequalities, graph solutions on number lines, and understand relationship quantities. Practice examples provided for better comprehension.

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Core Focus on Rational Numbers & Equations

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  1. Lesson 1.8 Core Focus onRational Numbers & Equations Linear Inequalities in One Variable

  2. Warm-Up Solve each equation for the variable. 1. 2. 3. h = 4 y = 2 x = 28

  3. Lesson 1.8 Linear Inequalities Solve inequalities with one variable.

  4. Vocabulary Inequality A mathematical sentence that contains < , >,  , or  to show a relationship between quantities. Nathan has more than $10 in his wallet. n > $10.00 Jackie has run at most 200 miles. j ≤ 200 Good to Know! Inequalities have multiple answers that can make the statement true. In Nathan’s example, he might have $20 or $100, all that is known for certain is that he has more than $10 in his wallet. There are an infinite number of possibilities that make the statement n > $10.00 true.

  5. Inequality Symbols > “greater than” < “less than”  “greater than or equal to”  “less than or equal to”

  6. Example 1 Write an inequality for each statement. a. Carla’s weight (w) is greater than 100 pounds. The key words are “greater than.” w > 100 b. Vicky has at most $500 in her savings account. Let m represent the amount of money in Vicky’s account. The key words are “at most.” This means m ≤ $500 she has less than or equal to $500. c. Quinton’s age is greater than 40 years old. Let a represent Quinton’s age. The key words are “greater than.” a > 40

  7. Extra Example 1 Write an inequality for each statement. a. Sharon earns at least 8 dollars (d) per baby-sitting job. b. Kenny does less than 10 hours (h) of homework per week. c. Rayanna is more than 48 inches (i) tall. d ≥ 8 h < 10 i> 48

  8. Graphing Inequalities 4 4 4 4 2 2 2 2 -2 -2 -2 -2 1 1 1 1 5 5 5 5 3 3 3 3 -4 -4 -4 -4 -5 -5 -5 -5 -1 -1 -1 -1 0 0 0 0 -3 -3 -3 -3 Solutions to an inequality can be graphed on a number line. For < or >, use an OPEN circle to graph the inequality: x>1 x<1 For  or , use a CLOSED circle to graph the inequality: x  1 x 1

  9. Example 2 8 6 2 5 9 7 -1 0 3 4 1 Inequalities are solved using properties similar to those you used to solve equations. Solve the inequality and graph its solution on a number line. Subtract 2 from both sides of the inequality. Multiply both sides of the inequality by 4. Graph the solution on a number line. Use a closed circle. – 2 – 2  4 4 

  10. Extra Example 2 Solve the inequality and graph its solution on a number line. 4x + 5 > 21 x > 4

  11. Example 3 Solve the inequality and graph its solution on a number line. 6x + 3 < 2x – 5 Subtract 2x from each side of the inequality. Subtract 3 from each side. Divide both sides by 4. Graph the solution on a number line. Use an open circle. 6x + 3 < 2x – 5 –2x –2x a 4x + 3< –5 a –3 –3 a 4x < –8 a 4 4a x < –2 a 3 1 -3 0 4 2 -6 -5 -2 -1 -4

  12. Extra Example 3 Solve the inequality and graph its solution on a number line. 2x − 5 ≤ 4x − 7 x ≥ 1

  13. Example 4 Solve the inequality.  4x + 7  19 Subtract 7 from each side of the inequality. Divide both sides by  4. Since both sides were divided by a negative, flip the inequality symbol. 4x + 7  19 7 7 4x  12 4 4 x  3  The sign changed direction because both sides were divided by a negative number.

  14. Extra Example 4 Solve the inequality 9 ≥ −3x + 15. x ≥ 2

  15. Communication Prompt Number lines are used to give a visual picture of an inequality statement. What is another situation in math where a visual is used to show math?

  16. Exit Problems 1. Write an inequality for the graph. 2. Write an inequality for the statement, “Lance walked more than 2 miles (m).” 3. Solve the inequality and graph the solution: 2x + 7 < 3. x≥ –3 m > 2 x < –2 1 1 -3 -3 0 0 2 2 -6 -6 -5 -5 -2 -2 -1 -1 -4 -4

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