Grade 6-Solving Inequalities

Pink= Tips and tricks — extra advice, bonus info, memory hacks

Blue = Answers — actual answers you need to memorize, solve problems, or short explanations.

Yellow =Super important stuff — definitions, key facts, dates, formulas.

Green= The Chapters

Chapter Overview

• Goal: Solve inequalities involving two operations and whole numbers up to 100.

Understanding Inequalities

• Definition: Inequalities are relationships between two expressions or values that are not equal.

• Signs Used:

  • Less than: (<)

  • Greater than: (>)

  • Less than or equal to: $(≤)$ -the “Aligator” sign is the less part and the line is Equal

  • Greater than or equal to: $(≥)$-the “Aligator” sign is the More part and the line is Equal

  • Not equal: $(≠)$ (we might use this in the test)

• Examples:

  • (a < b): a is less than b

  • (a > b): a is greater than b

  • $(a ≠ b)$: a is not equal to b

Common Words Expressing Inequalities

• Greater than: More than, Exceeding, Above

• Less than: Fewer than, Below

• At least: No fewer than, Minimum

• Not above: Does not exceed, At most

Solving Inequalities

• Approach: Treat like solving equations. Keep the unknown variable on one side.

• Balance Model:

  • Example: Solve 4a + 3 < 23

    • Subtract 3 from both sides:
      4a < 20

      Divide by 4:
      a < 5

  • Note: If the inequality sign changes, the direction of the inequality also needs to be reversed.

    4a+3<23

    4a-3<20 =4a<20$</p><p>$=4a<20$</p><p>$\frac44$(you have to do this on the 4 x a or 4a,this whould)$4a < 20

Practicing Inequalities

• Solve the following inequalities:

  • a) 3a + 5 < 23$</p></li><li><p>b)$α - 3 > 9$</p></li><li><p>c)$2a + a - 1 > 35$</p></li><li><p>d)$7a + a + 8 < 80

Graphing Inequalities

• Number Line: Shows the range of true values for an inequality.

• Symbols Used:

  • Open dot: used for <$or$>$denote values not included.</p></li><li><p>Closed dot: used for$≤$or$≥$indicating values included.</p></li></ul></li><li><p><strong>Example</strong>:</p><ul><li><p>Graphing$x < 4$and$x ≥ 4 shows respective points and their meanings.

Example: Solving a Real-Life Inequality

• Scenario: Nadia has $100 and needs at least $300 for a gift. Each babysitting job earns her $20.

• Problem Setup: Let d$be the number of days.</p></li><li><p><strong>Inequality Formulation</strong>:</p><ul><li><p>Equation:$100 + 20d ≥ 300$</p></li><li><p>Solve:</p><ul><li><p>$20d ≥ 200$</p></li><li><p>$d ≥ 10

Another Example: Taxi Fare Calculation

• Problem: Kerry needs to know how far he can travel with $50 if the taxi charges $2 flat and $0.50/km.

• Inequality Formulation:

  • Setup: 2 + 0.50x ≤ 50

  • Solve for x$$, then graph the result on a number line to illustrate possible distances.

Conclusion

• Final Reminder: Make sure to complete and turn in all work via Google Classroom!