4.3 Solving Linear Equations & Inequalities

4.3 Solving Linear Equations & Inequalities

Conceptual Foundation: The Balance Model & Inverse Operations

Solving linear equations and inequalities is a core expectation of Ontario middle-school and secondary mathematics curriculum. On the MPT, candidate teachers must demonstrate procedural accuracy, conceptual understanding of equation balance, and mastery of algebraic properties.

An equation is a mathematical statement asserting that two expressions are equal in value ($LHS = RHS$). The balance scale model provides the foundational intuition: an equation is like a balanced two-pan scale. To maintain balance, any mathematical operation performed on the left-hand side (LHS) must be identically performed on the right-hand side (RHS).

Inverse Operations

To isolate a variable, we perform inverse operations in reverse order of operations (reverse BEDMAS):

• Addition ($\mathbf{+}$) is inverted by Subtraction ($\mathbf{-}$)
• Subtraction ($\mathbf{-}$) is inverted by Addition ($\mathbf{+}$)
• Multiplication ($\mathbf{\times}$) is inverted by Division ($\mathbf{\div}$)
• Division ($\mathbf{\div}$) is inverted by Multiplication ($\mathbf{\times}$)

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Solving 1-Step and 2-Step Linear Equations

1-Step Equations

A single inverse operation isolates the variable.

• Example: $x + 14 = 31 \implies x = 31 - 14 \implies x = 17$
• Example: $-6x = 42 \implies x = \frac{42}{-6} \implies x = -7$

2-Step Equations

Requires undoing addition/subtraction first, followed by multiplication/division.

$$\text{Solve: } 4x - 9 = 27$$

1. Add 9 to both sides: $4x = 27 + 9 \implies 4x = 36$
2. Divide both sides by 4: $x = \frac{36}{4} \implies x = 9$

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Multi-Step Equations with Variables on Both Sides

When variables appear on both sides of the equals sign, use the following systematic 5-step algorithm:

Systematic Solving Algorithm:

1. Clear Parentheses: Apply the distributive property to expand all brackets.
2. Simplify Each Side: Combine like terms independently on the LHS and RHS.
3. Collect Variable Terms: Add or subtract variable terms to move all variable terms to one side of the equation.
4. Collect Constant Terms: Add or subtract constants to move all numerical terms to the opposite side.
5. Isolate the Variable: Divide or multiply by the variable's coefficient to solve for $x$.

Example Worked Problem:

$$\text{Solve: } 5(x - 2) + 4 = 2(2x + 7) - 3$$

• Step 1 (Expand): $5x - 10 + 4 = 4x + 14 - 3$
• Step 2 (Combine like terms): $5x - 6 = 4x + 11$
• Step 3 (Collect variables on LHS): Subtract $4x$ from both sides: $$5x - 4x - 6 = 11 \implies x - 6 = 11$$
• Step 4 (Collect constants on RHS): Add 6 to both sides: $$x = 11 + 6 \implies x = 17$$
• Step 5 (Verification): Substitute $x = 17$ into original equation:
  • $\text{LHS} = 5(17 - 2) + 4 = 5(15) + 4 = 75 + 4 = 79$
  • $\text{RHS} = 2(2(17) + 7) - 3 = 2(34 + 7) - 3 = 2(41) - 3 = 82 - 3 = 79$
  • Since $\text{LHS} = \text{RHS} = 79$, the solution $x = 17$ is verified.

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Clearing Fractional Denominators Using LCD

Equations containing fractions are a frequent source of error on the MPT. The most efficient strategy is to clear all fractions in Step 1 by multiplying every term on both sides of the equation by the Least Common Denominator (LCD) of all denominators present.

Step-by-Step LCD Fraction Clearing Technique:

Given:

$$\frac{2x + 1}{3} - \frac{x - 2}{4} = 2$$

1. Identify the LCD: Denominators are 3 and 4. The LCD is 12.
2. Multiply EVERY term by 12: $$12 \cdot \left(\frac{2x + 1}{3}\right) - 12 \cdot \left(\frac{x - 2}{4}\right) = 12 \cdot 2$$
3. Simplify numerator multipliers: $$4(2x + 1) - 3(x - 2) = 24$$
4. Expand brackets: $$8x + 4 - 3x + 6 = 24$$
5. Combine like terms: $$5x + 10 = 24$$
6. Isolate variable: $$5x = 24 - 10 \implies 5x = 14 \implies x = \frac{14}{5} = 2.8$$

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Solving Linear Inequalities & The Negative Division Rule

A linear inequality uses order symbols ($<, \le, >, \ge$) instead of an equals sign. The solution set represents a range of values rather than a single number.

Rules of Inequality Manipulation

All equation balance operations apply to inequalities with one vital exception:

The Golden Rule of Inequalities: When you multiply or divide both sides of an inequality by a negative number, you MUST reverse (flip) the direction of the inequality sign.

Mathematical Proof of Sign Flipping:

Consider the true statement: $2 < 5$.

• If we multiply both sides by $-1$:
  • Left side becomes $-2$. Right side becomes $-5$.
  • On a number line, $-2$ is to the right of $-5$, so $-2 > -5$.
  • Maintaining $2 < 5 \implies -2 < -5$ would be mathematically false! Thus, the sign must reverse.

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Step-by-Step Worked Equation & Inequality Solutions

Problem 1: Equation with Decimals and Distribution

Problem: Solve for $x$: $0.4(2x - 5) = 1.2x - 3.6$

Step-by-Step Solution:

• Option A: Clear decimals by multiplying entire equation by 10. $$10 \cdot [0.4(2x - 5)] = 10 \cdot (1.2x - 3.6)$$ $$4(2x - 5) = 12x - 36$$
• Step 2: Expand left side. $$8x - 20 = 12x - 36$$
• Step 3: Collect variable terms on right side. $$-20 = 12x - 8x - 36 \implies -20 = 4x - 36$$
• Step 4: Add 36 to both sides. $$16 = 4x$$
• Step 5: Divide by 4. $$x = 4$$

Problem 2: Multi-Step Inequality with Fraction & Negative Coefficient

Problem: Solve the inequality for $x$: $\frac{7 - 3x}{4} \le 4$

Step-by-Step Solution:

• Step 1: Multiply both sides by $+4$ (positive, so sign does NOT flip). $$7 - 3x \le 4 \cdot 4$$ $$7 - 3x \le 16$$
• Step 2: Subtract 7 from both sides. $$-3x \le 16 - 7$$ $$-3x \le 9$$
• Step 3: Divide both sides by $-3$. (DIVIDING BY NEGATIVE: FLIP SIGN from $\le$ to $\ge$). $$x \ge \frac{9}{-3}$$ $$x \ge -3$$
• Step 4: Conclusion. The solution set is all real numbers greater than or equal to $-3$.