Unit 1 - Lesson 5: Solving Linear Equations and Inequalities in One Variable in R with graphical representations
Solving first-degree linear equations and inequalities in one variable within the real number system R represents the prime algebraic application connecting variable balance with continuous interval representations. Equations rely on field balance axioms to isolate the unknown variable, extending to equations involving quadratic and cubic powers whose roots map directly to real solutions. Inequalities, on the other hand, demand rigorous adherence to order axioms, most notably the cardinal rule requiring the reversal of the inequality sign upon multiplying or dividing by a negative real number. Solutions are formally expressed in interval notation and graphed on the real number line using precise hollow or solid boundary markers in strict accordance with national pedagogical criteria.
1. First-Degree Linear Equations in R
Standard form: ax + b = c (where a ≠ 0):
- Radical Coefficient Equations:
√2 x - 1 = 3 ⟹ √2 x = 4 ⟹ x = 4 / √2 = 2√2. - Cubic Power Equations:
x³ + 5 = 32 ⟹ x³ = 27 ⟹ x = ∛27 = 3. - Quadratic Equations:
x² = 9 ⟹ x = ±√9 = ±3(two real solutions).
2. Order Axioms & Inequality Sign Reversal
Rules governing operations on inequalities (>, <, ≥, ≤):
| Operation | Effect on Inequality Sign | Example |
|---|---|---|
| Adding / Subtracting | Sign remains unchanged | x - 2 > 5 ⟹ x > 7 |
| Multiplying / Dividing by Positive | Sign remains unchanged | 2x < 6 ⟹ x < 3 |
| Multiplying / Dividing by Negative | Sign must be reversed immediately! | -3x ≤ 9 ⟹ x ≥ -3 |
Warning: Forgetting to reverse the inequality sign when dividing by a negative number is a fatal algebraic error.
3. Compound / Double Linear Inequalities
Inequalities with three parts: a < bx + c ≤ d:
- Step 1: Subtract constant
cfrom all three parts:(a - c) < bx ≤ (d - c). - Step 2: Divide all three parts by coefficient
b. - Step 3: Write the solution set as a bounded interval and graph on the real number line.
- Example:
-1 < 2x + 3 ≤ 7 ⟹ -4 < 2x ≤ 4 ⟹ -2 < x ≤ 2 ⟹ S.S. = (-2, 2].
4. Graphical Representation on the Number Line
Standardized graphing conventions:
- Strict Inequalities (> or <): Unshaded (hollow) circle at the boundary coordinate.
- Non-strict Inequalities (≥ or ≤): Shaded (solid) circle at the boundary coordinate.
- Ray Direction: Rightwards towards
+∞if x > a, leftwards towards-∞if x < a.
Exercise 1: Radical Linear Equation
Problem: Solve in R: √3 x - 2 = 4.
• Add 2 to both sides: √3 x = 6.
• Divide by √3: x = 6 / √3.
• Rationalize: x = (6 × √3) / 3 = 2√3.
• Solution set S.S. = {2√3}.
Exercise 2: Simple Linear Inequality
Problem: Solve in R: 3x + 2 < 11, and graph the solution set.
• Subtract 2: 3x < 9.
• Divide by 3: x < 3.
• Interval notation: S.S. = (-∞, 3).
• Graph: Hollow circle at 3 with arrow extending leftwards to -∞.
Exercise 3: Inequality with Negative Coefficient
Problem: Solve in R: 5 - 2x ≤ 13.
• Subtract 5: -2x ≤ 8.
• Divide by -2 and reverse inequality sign: x ≥ -4.
• S.S. = [-4, ∞).
• Graph: Solid circle at -4 with arrow extending rightwards to ∞.
Exercise 4: Compound Double Inequality
Problem: Solve in R: 1 ≤ 2x - 3 < 5.
• Add 3 to all parts: 4 ≤ 2x < 8.
• Divide by 2: 2 ≤ x < 4.
• S.S. = [2, 4).
• Graph: Solid circle at 2, hollow circle at 4, shading between them.
Exercise 5: Cubic Polynomial Equation
Problem: Solve in R: (x - 1)³ + 10 = 2.
• Subtract 10: (x - 1)³ = -8.
• Cube root: x - 1 = ∛(-8) = -2.
• x = -2 + 1 = -1.
• S.S. = {-1}.
Exercise 6: Quadratic Equation in R
Problem: Solve in R: 2x² - 5 = 11.
• 2x² = 16 ⟹ x² = 8.
• x = ±√8 = ±2√2.
• S.S. = {2√2, -2√2}.
Exercise 7: Variables on Both Sides
Problem: Solve in R: 4x + 1 > 2x + 7.
• Collect like terms: 4x - 2x > 7 - 1.
• 2x > 6 ⟹ x > 3.
• S.S. = (3, ∞).
Exercise 8: Double Inequality with Negative Slope
Problem: Solve in R: -3 ≤ 1 - 2x < 5.
• Subtract 1: -4 ≤ -2x < 4.
• Divide by -2 and flip signs: 2 ≥ x > -2.
• Rewritten from least to greatest: -2 < x ≤ 2.
• S.S. = (-2, 2].
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