Linear Inequalities

Introduction to Inequalities

An inequality compares two quantities using symbols:

• $<$ (less than)
• $>$ (greater than)
• $\le$ (less than or equal to)
• $\ge$ (greater than or equal to)

Example:

• $x>3$, $2x-5\le7$

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Linear Inequalities in One Variable

A linear inequality in one variable is of the form: $$ax+b < c,\quad ax+b \le c,\quad ax+b > c,\quad ax+b \ge c$$ where $a\ne0$.

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Rules for Solving Inequalities

Rule 1: Addition/Subtraction Rule

If the same number is added to or subtracted from both sides, the inequality sign does not change.

Example: $x-3>5 \Rightarrow x>8$

Rule 2: Multiplication/Division Rule

If both sides are multiplied or divided by a positive number, the inequality sign remains the same.

If both sides are multiplied or divided by a negative number, the inequality sign reverses.

Example: $-2x<6 \Rightarrow x>-3$

Graphical Representation on Number Line

Solutions of inequalities are represented on the number line.

• Open circle (○): for $<$ or $>$
• Closed circle (●): for $\le$ or $\ge$

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Compound Inequalities

(A) AND type

Example: $$2<x<5$$ Solution: all real numbers between 2 and 5.

(B) OR type

Example: $$x<1 \text{ or } x>4$$ Solution: two separate intervals.

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Inequalities Involving Absolute Values

Case 1: $|x|<a$

Equivalent to: $$-a<x<a$$

Case 2: $|x|>a$

Equivalent to: $$x<-a \text{ or } x>a$$

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Solution Set and Interval Notation

Inequality	Interval Form
$x>a$	$(a,\infty)$
$x\ge a$	$[a,\infty)$
$x<a$	$(-\infty,a)$
$x\le a$	$(-\infty,a]$

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Common Board Mistakes

1. Forgetting to reverse inequality sign when multiplying/dividing by a negative number.
2. Confusing open and closed circles on number line.
3. Writing incorrect interval notation.
4. Solving absolute value inequalities like equations.
5. Ignoring domain restrictions.

Solved Examples (15+) — Linear Inequalities

Example 1

Solve: $x+3>7$.

Solution: $x>4$.

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Example 2

Solve: $2x-5\le9$.

Solution: $2x\le14 \Rightarrow x\le7$.

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Example 3

Solve: $5x>20$.

Solution: $x>4$.

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Example 4

Solve: $-3x<12$.

Solution: Divide by $-3$ and reverse sign: $x>-4$.

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Example 5

Solve: $4-2x\ge0$.

Solution: $-2x\ge-4 \Rightarrow x\le2$.

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Example 6

Solve: $\frac{x}{3}-2>1$.

Solution: $\frac{x}{3}>3 \Rightarrow x>9$.

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Example 7

Solve: $2(3x-1)\le10$.

Solution: $6x-2\le10 \Rightarrow 6x\le12 \Rightarrow x\le2$.

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Example 8

Solve: $-4(2x+1)>8$.

Solution: $-8x-4>8 \Rightarrow -8x>12 \Rightarrow x<-\frac{3}{2}$.

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Example 9

Solve: $1<2x+3<9$.

Solution: $-2<2x<6 \Rightarrow -1<x<3$.

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Example 10

Solve: $|x|<5$.

Solution: $-5<x<5$.

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Example 11

Solve: $|x|\ge4$.

Solution: $x\le-4$ or $x\ge4$.

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Example 12

Solve: $|2x-1|<3$.

Solution: $-3<2x-1<3$ $-2<2x<4$ $-1<x<2$.

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Example 13

Solve: $|3x+2|>5$.

Solution: $3x+2>5$ or $3x+2<-5$ $x>1$ or $x<-\frac{7}{3}$.

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Example 14

Write solution of $x\ge-2$ in interval notation.

Solution: $[-2,\infty)$.

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Example 15

Represent $x<3$ on number line.

Solution: Open circle at 3 and shading to the left.

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Example 16

Solve: $2x+5<3x-1$.

Solution: $6<x \Rightarrow x>6$.

Questions and Answers (Board Exam)

Q1. State the rules for solving linear inequalities.

Answer:

• Adding/subtracting same number keeps inequality unchanged.
• Multiplying/dividing by positive number keeps inequality unchanged.
• Multiplying/dividing by negative number reverses inequality sign.

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Q2. How is $x<5$ represented on number line?

Answer: Open circle at 5 with shading towards left.

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Q3. Solve $|x|<a$.

Answer: $-a<x<a$.

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Q4. Solve $|x|>a$.

Answer: $x<-a$ or $x>a$.

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Q5. Write interval notation for $-3<x\le2$.

Answer: $(-3,2]$.