一元一次不等式yīyuán yīcì bùděngshì

Core Concept

A linear inequality in one variable is an inequality that can be written in the form:

$ax + b > 0, \quad ax + b < 0, \quad ax + b \geq 0, \quad \text{or} \quad ax + b \leq 0$

where $a \neq 0$ and $a$, $b$ are constants.

Standard Forms

• $ax + b > 0$ (strictly greater than)
• $ax + b < 0$ (strictly less than)
• $ax + b \geq 0$ (greater than or equal to)
• $ax + b \leq 0$ (less than or equal to)

Properties of Inequalities

1. Addition/Subtraction Property

If $a > b$, then $a + c > b + c$ for any real number $c$.

Adding or subtracting the same number from both sides preserves the inequality direction.

2. Multiplication/Division Property

Positive multiplier: If $a > b$ and $c > 0$, then $ac > bc$.

Negative multiplier: If $a > b$ and $c < 0$, then $ac < bc$.

Critical Rule: When multiplying or dividing by a negative number, reverse the inequality sign.

3. Transitivity

If $a > b$ and $b > c$, then $a > c$.

Solution Method

Step-by-Step Process

1. Simplify both sides (distribute, combine like terms)
2. Move variable terms to one side
3. Move constant terms to the other side
4. Divide by the coefficient of $x$
5. If dividing by negative, flip the inequality sign
6. Express solution in interval notation

Worked Example

Solve: $3x - 7 \leq 2x + 5$

Step 1: Subtract $2x$ from both sides $3x - 2x - 7 \leq 5$ $x - 7 \leq 5$

Step 2: Add 7 to both sides $x \leq 12$

Solution: $(-\infty, 12]$

Examples with Sign Reversal

Example: Negative Coefficient

Solve: $-2x + 6 > 10$

Step 1: Subtract 6 $-2x > 4$

Step 2: Divide by -2 (reverse the sign!) $x < -2$

Solution: $(-\infty, -2)$

CSCA Practice Problems

💡 Note: The following practice problems are designed based on the CSCA exam syllabus.

Example 1: Basic (Difficulty ★☆☆☆☆)

Solve $2x - 5 > 7$.

Options:

• A. $x > 6$
• B. $x < 6$
• C. $x > 1$
• D. $x < 1$

Solution: $2x - 5 > 7$ $2x > 12$ $x > 6$

Answer: A

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Example 2: Intermediate (Difficulty ★★★☆☆)

Solve $\dfrac{2x - 1}{3} - \dfrac{x + 2}{2} \geq 1$.

Solution:

Multiply by LCD = 6: $2(2x - 1) - 3(x + 2) \geq 6$ $4x - 2 - 3x - 6 \geq 6$ $x - 8 \geq 6$ $x \geq 14$

Answer: $[14, +\infty)$

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Example 3: System of Inequalities (Difficulty ★★★☆☆)

Find the solution set of the system: $\begin{cases} 2x - 1 > 3 \\ x + 4 \leq 10 \end{cases}$

Solution:

From first inequality: $2x > 4$, so $x > 2$ From second inequality: $x \leq 6$

Combined: $2 < x \leq 6$

Answer: $(2, 6]$

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Example 4: Advanced (Difficulty ★★★★☆)

For what values of $a$ does $ax - 2 > ax + 1$ have no solution?

Solution:

Subtract $ax$ from both sides: $-2 > 1$

This is always false, regardless of $a$.

Therefore, the inequality has no solution for all real values of $a$.

Answer: $a \in \mathbb{R}$ (all real numbers)

Compound Inequalities

"And" Type (Intersection)

$a < x$ AND $x < b$ means $a < x < b$

Example: $-3 < 2x + 1 < 7$

Solve as: $-3 < 2x + 1$ AND $2x + 1 < 7$

• $-4 < 2x$ → $x > -2$
• $2x < 6$ → $x < 3$

Solution: $-2 < x < 3$, or $(-2, 3)$

"Or" Type (Union)

$x < a$ OR $x > b$

Example: $3x - 2 < 1$ OR $3x - 2 > 7$

Solve each:

• $3x < 3$ → $x < 1$
• $3x > 9$ → $x > 3$

Solution: $x < 1$ or $x > 3$, or $(-\infty, 1) \cup (3, +\infty)$

Common Mistakes

❌ Mistake 1: Forgetting to Flip the Sign

Wrong: $-3x > 6 \Rightarrow x > -2$ ✗

Correct: $-3x > 6 \Rightarrow x < -2$ ✓

❌ Mistake 2: Incorrect Fraction Operations

Wrong: Multiply $\dfrac{x}{2} > 3$ by 2 to get $x > 1.5$ ✗

Correct: $\dfrac{x}{2} > 3 \Rightarrow x > 6$ ✓

❌ Mistake 3: Wrong Interval Notation

Wrong: $x > 3$ means $(3, \infty]$ ✗

Correct: $x > 3$ means $(3, +\infty)$ ✓

Study Tips

1. ✅ Mark inequality direction: Keep track during each step
2. ✅ Highlight negative multipliers: These require sign reversal
3. ✅ Check with test values: Verify your answer makes the original true
4. ✅ Draw number line: Visualize the solution set

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💡 Exam Tip: The most common error is forgetting to reverse the inequality when dividing by a negative number. Always highlight this step!