Grade 9 Mathematics - Unit 6: Linear Equations and Inequalities

6.1 Solving Equations Using Inverse Operations

  • Inverse operations "undo" or reverse each other's results.
  • Inverse operations take us back to where we started.
    • Example: 2+3=52 + 3 = 5 and 5−3=25 - 3 = 2
    • 2×3=62 × 3 = 6 and 63=2\frac{6}{3} = 2
  • To solve equations:
    • Determine the operations applied to the variable.
    • Use inverse operations to isolate the variable.
    • Whatever you do to one side, do to the other to keep the equation balanced.
    • Example: To solve x+2.4=6.5x + 2.4 = 6.5

Solving Two-Step Equations

  • When more than one operation acts on a variable, reverse the order of operations.
    1. Reverse addition and subtraction outside brackets.
    2. Reverse multiplication and division outside brackets.
    3. Remove outermost brackets and reverse operations in order.
  • To “undo” a sequence of operations, perform inverse operations in reverse order.
  • Examples:
    • Solve 5x+9=445x + 9 = 44
      • Reverse addition: Subtract 9 from both sides: 5x=355x = 35
      • Reverse multiplication: Divide both sides by 5: x=7x = 7
      • Verify: 5(7)+9=445(7) + 9 = 44
    • Solve 4.5d−3.2=−18.54.5d - 3.2 = -18.5
      • Reverse subtraction: Add 3.2 to both sides: 4.5d=−15.34.5d = -15.3
      • Reverse multiplication: Divide both sides by 4.5: d=−3.4d = -3.4
      • Verify: 4.5(−3.4)−3.2=−18.54.5(-3.4) - 3.2 = -18.5
    • Solve x4+3=7.2\frac{x}{4} + 3 = 7.2
      • Reverse addition: Subtract 3 from both sides: x4=4.2\frac{x}{4} = 4.2
      • Reverse division: Multiply both sides by 4: x=16.8x = 16.8
      • Verify: 16.84+3=7.2\frac{16.8}{4} + 3 = 7.2
    • Solve 3(y4)−1=153(\frac{y}{4}) - 1 = 15
      • Reverse subtraction: Add 1 to both sides: 3(y4)=163(\frac{y}{4}) = 16
      • Reverse multiplication: Divide by 3: y4=163\frac{y}{4} = \frac{16}{3}
      • Reverse division: Multiply by 4: y=643y = \frac{64}{3}
      • Verify: 3(64/34)−1=153(\frac{64/3}{4}) - 1 = 15
    • Solve 43(z−11)+6=48\frac{4}{3}(z - 11) + 6 = 48
      • Reverse addition: Subtract 6: 43(z−11)=42\frac{4}{3}(z - 11) = 42
      • Reverse multiplication: Multiply by 34\frac{3}{4}: z−11=632z - 11 = \frac{63}{2}
      • Reverse subtraction: Add 11: z=632+11z = \frac{63}{2} + 11
      • Simplify: z=852z = \frac{85}{2}
      • Verify: 43(852−11)+6=48\frac{4}{3}(\frac{85}{2} - 11) + 6 = 48
  • Simplify the equation before reversing operations.
    • Example: Solve 6x−5−2x+3−2=46x - 5 - 2x + 3 - 2 = 4
      • Simplify: 4x−4=44x - 4 = 4
      • Reverse subtraction: Add 4 to both sides: 4x=84x = 8
      • Reverse multiplication: Divide both sides by 4: x=2x = 2
      • Verify: 6(2)−5−2(2)+3−2=46(2) - 5 - 2(2) + 3 - 2 = 4
  • Word Problems:
    • Seven percent of a number is 56.7. Find the number.
      • Equation: 0.07x=56.70.07x = 56.7
      • Solution: x=810x= 810
    • Rectangle with length 3.7 cm and perimeter 13.2 cm. Find the width.
      • Equation: 2(3.7)+2w=13.22(3.7) + 2w = 13.2

Solving Equations Involving the Distributive Property

  • Examples:
    • 2(3.7+x)=13.22(3.7 + x) = 13.2
    • 6=1.5(x−6)6 = 1.5(x - 6)
    • 3(x−5)=23(x - 5) = 2

6.2 Solving Equations Using Balance Strategies

  • Isolate the variable.
  • Inverse operations only work when the variable appears once.
  • Balance strategy: Whatever is done to one side, do the same to the other to keep the equation balanced.
  • Get the variable on one side and the constant term on the other.
  • Modeling Equations with Variable on Both Sides:
    • Example: 6x+2=10+4x6x + 2 = 10 + 4x
  • Balance Scale Model Limitations:
    • Cannot use when any term in the equation is negative.
  • Solving Equations Using Algebra Tiles:
    • Isolate the variable tiles and identify zero pairs.
    • Example: −2x+4=x−2-2x + 4 = x - 2
    • Inefficient for large numbers, fractions, and decimals.

Solving Equations with Variables on Both Sides

  • Goal: Get the variable on one side and the numbers on the other side.
  • Examples:
    • 7.4+4x=2x+13.27.4 + 4x = 2x + 13.2
    • 4x+7=21−3x4x + 7 = 21 - 3x
    • 3x+3=5x−53x + 3 = 5x - 5

Solving Equations with Brackets

  • Examples:
    • 1.5(x−6)=61.5(x - 6) = 6
    • 3(x−5)=23(x - 5) = 2
    • 4(x−5)=−2(x−2)4(x - 5) = -2(x - 2)
    • 3(x+1)=5(x−1)3(x + 1) = 5(x - 1)
  • Word Problem: Taxi Companies
    • Company A: $3.00 + $0.20 per km
    • Company B: $2.50 + $0.25 per km
    • At what distance will the cost be the same?
      • Equation: 3+0.20x=2.50+0.25x3 + 0.20x = 2.50 + 0.25x

Solving Equations with Fractions

  • Eliminate denominators by multiplying each term by a common denominator.
  • Examples:
    • x3=3\frac{x}{3} = 3
    • x4+x2=x5+7\frac{x}{4} + \frac{x}{2} = \frac{x}{5} + 7
    • x4+52=53\frac{x}{4} + \frac{5}{2} = \frac{5}{3}
    • 53(x−1)=2+(1−x)\frac{5}{3}(x - 1) = 2 + (1 - x)
    • (x+5)3=(x+5)4\frac{(x + 5)}{3} = \frac{(x + 5)}{4}

6.3 Introduction to Linear Inequalities

  • Model situations with a range of numbers instead of a single number.
    • Symbols: > (greater than), ≥ (greater than or equal to), < (less than), ≤ (less than or equal to)
  • Example: Time, tt, for which a car could be legally parked: t > 30, t≥30t ≥ 30, t < 30, t≤30t ≤ 30

Writing an Inequality to Describe a Situation

  • Examples:
    • Contest entrants must be at least 18 years old: Let aa be the age, a≥18a ≥ 18
    • Temperature has been below −5-5°C for the last week: Let TT be the temperature, T < -5
    • You must have 7 items or less to use the express checkout line: Let ii be the number of items, i≤7i ≤ 7
    • Scientists have identified over 400 species of dinosaurs: Let ss be the number of species, s > 400

Determining Whether a Number is a Solution of an Inequality

  • A linear equation is true for only one value.
  • A linear inequality may be true for many values.
  • The solution of an inequality is any value that makes the inequality true.
  • Example: b≥3b ≥ 3
  • Example: Is each number a solution to the inequality x > -2?
    • −8-8 (No)
    • −2-2 (No)
    • 00 (Yes)
    • 22 (Yes)
    • −2.5-2.5 (No)
    • −3.5-3.5 (No)

Graphing Inequalities on a Number Line

  • Show solutions on a number line.
  • Examples:
    • t > -5
    • −2≥x-2 ≥ x
    • 0.5≤a0.5 ≤ a
    • p < \frac{3}{2}

Describing Situations with Inequalities and Graphs

  • Examples:
    • A number is greater than 5: x > 5
    • A number is less than or equal to 4: x≤4x ≤ 4
    • The temperature is below −1-1°C today: T < -1
    • You have to be at least 16 years old to drive: a≥16a ≥ 16
    • You must have 10 items or less to go to the express lane: i≤10i ≤ 10 (Discrete data; whole numbers only)
    • Sarah’s mom said she should invite at least 10 people to her birthday party.
  • Summary When graphing inequalities:
    • > or < use hollow dots on the number line
    • ≥ or ≤ use solid dots on the number line
    • if continuous data, shade the line
    • if discrete data, use dots on whole numbers

Additional Examples

  • Write the inequality for each of the following number lines.

6.4 Solving Linear Inequalities Using Addition & Subtraction

Let’s Investigate!!

Part A: Add a Positive and Negative Number
  • Do the inequalities hold true?
Part B: Subtract a Positive and Negative Number
  • Do the inequalities hold true?

Examples

  • Solve each of the following inequalities, verify the solution and graph on a number line.
    • 6.2≤x−4.56.2 ≤ x - 4.5
    • 4.8 + x > -3.2
    • x + 4 < 7

Word Problem Example

  • Jack plans to board his dog while he is away on vacation.
    • Boarding House A charges $90 plus $5 per day
    • Boarding House B charges $100 plus $4 per day
  • For how many days must Jack board his dog in House A to be less expensive than House B?
    • Choose a variable and write an inequality.
    • Solve the problem.
    • Graph the solution.

6.5 – Solving Linear Inequalities Using Multiplication & Division

Let’s Continue to Investigate!!

Part C: Multiply by a Positive and Negative Number
  • Do the inequalities hold true?
Part D: Divide by a Positive and Negative Number
  • Do the inequalities hold true?

Examples

  • Solve each of the following inequalities, verify the solution and graph on a number line.
    • −5x≤25-5x ≤ 25
    • 7a < -21
    • \frac{a}{4} > -3
  • Solve and verify: -2.6x + 14.6 > -5.2 + 1.8x

Word Problem

  • A super slide charges $1.25 to rent a mat and $0.75 per ride. James has $10.25. How many rides can James go on?
    • Choose a variable and write an inequality.
    • Solve the problem.
    • Graph the solution.

Summary for Solving Inequalities

  • Rules for Solving Inequalities:
    1. Make the same changes to both sides of the inequality
    2. Isolate the variable
    3. Combine Like Terms
    4. Use the Inverse Operation to remove clutter away from variable
    5. BUT, if your Inverse Operation is multiplication or division by a negative number, the inequality sign reverses
      • < becomes >
      • > becomes <
      • ≤ becomes ≥
      • ≥ becomes ≤

Solving Two-Step Inequalities

  1. Add or subtract to isolate the variable term.
  2. Multiply or divide to solve for the variable. If multiply or divide by a negative number then reverse the inequality symbol.
    • Example:
      • −3x+5≤−16-3x+5≤-16
        • −5−5-5 -5
        • Subtract
        • −3x≤−21-3x≤-21
        • −3x−3≥−21−3\frac{-3x}{-3} ≥ \frac{-21}{-3}
        • Divide by -3, reverse inequality
        • x≥7x ≥ 7

Inequalities vs. Equations

  • An equation has only one solution, while an inequality has a range of solutions.

Extra Examples

  • Solve each inequality and graph the solution on a number line.
    • 3x + 1 > 10
    • −3(x−2)≤9-3(x - 2) ≤ 9
    • 3x + 1 > 4x - 2
    • 3x≥32(x−2)3x ≥ \frac{3}{2}(x - 2)

Word Problems: Linear Equations & Inequalities

  1. Write an equation or inequality for each statement below and solve. Show the solutions for all inequalities on a number line.
    • Triple a number decreased by one is less than 11.
    • A number multiplied by 4, increased by 5 is 1.
    • You can invite at most 5 friends over to your house Saturday evening.
    • Five subtract 3 times a number is equal to 3.5 times the same number subtract eight.
    • Sam has a choice of two companies to rent a car. Company A charges $199 per week plus $0.20 per km driven. Company B charges $149 per week plus $0.25 per km driven. At what distance will both companies cost the same?
  2. For the inequality 25−3x≥−7x+2\frac{2}{5} - 3x ≥ -7x + 2, Kate says the solution is x≥−8x ≥ -8. Choose values to verify whether or not this is correct.
  3. Michael is 7 years younger than his sister, Sarah. How old must each be if the sum of their ages is greater than 25?
  4. Jenna rents a car for $350 plus $12.50 per day on her vacation. If she budgeted $900 for her car rental, for how many days can she rent the car? Graph the solution on a number line.