Interval Notation and Solving Linear Inequalities

Interval Notation

Introduction to Interval Notation

Interval notation is a concise way to represent sets of numbers, especially solution sets for inequalities. It is often used instead of set notation to describe all numbers between two endpoints.

• Set Notation: Describes a set using braces and a condition, e.g., { x | 1 ≤ x ≤ 5 }.

• Interval Notation: Uses parentheses and brackets to indicate endpoints and whether they are included or excluded.

Types of Intervals

• Closed Interval [a, b]: Both endpoints are included. Example: $[1, 5]$ means $1 \leq x \leq 5$.

• Open Interval (a, b): Both endpoints are excluded. Example: $(0, 5)$ means $0 < x < 5$.

• Half-Open (Half-Closed) Interval: One endpoint is included, the other is not. Example: $(0, 5]$ means $0 < x \leq 5$.

Type	Set Notation	Interval Notation	Number Line	Endpoints
Closed	{ x | a ≤ x ≤ b }	[a, b]	a ●-----● b	Included
Open	{ x | a < x < b }	(a, b)	a ○-----○ b	Excluded
Half-Open	{ x | a < x ≤ b }	(a, b]	a ○-----● b	One included, one excluded

Infinity in Interval Notation

• When an interval extends indefinitely, use $\infty$ (infinity) or $-\infty$ (negative infinity).

• Infinity symbols are always paired with parentheses, never brackets, because infinity is not a specific number.

• Example: $(-\infty, 0]$ means all real numbers less than or equal to 0.

Examples

• Set: { x | x ≥ 3 } Interval Notation: $[3, \infty)$

• Set: { x | 0 < x < 5 } Interval Notation: $(0, 5)$

Solving Linear Inequalities

Introduction to Linear Inequalities

Linear inequalities are similar to linear equations but use inequality symbols (<, ≤, >, ≥) instead of an equal sign. The solution is a range of values rather than a single value.

• Linear Equation: $2x - 6 = 0$

• Linear Inequality: $2x - 6 < 0$

Solving Steps

1. Solve as you would a linear equation: isolate the variable using addition, subtraction, multiplication, or division.

2. Important: When multiplying or dividing both sides by a negative number, reverse the direction of the inequality symbol.

Example: Solving a Linear Inequality

• Solve $2x - 6 < 0$:

• $2x < 6$

• $x < 3$

• Interval Notation: $(-\infty, 3)$

Example: Multiplying/Dividing by a Negative

• Solve $-2x < 6$:

• Divide both sides by -2 (reverse the inequality):

• $x > -3$

• Interval Notation: $(-3, \infty)$

Practice Problems

• Problem: $x | 1 \leq x \leq 26$ Interval Notation: $[1, 26]$

• Problem: $x | x \leq 7$ Interval Notation: $(-\infty, 7]$

Fractions & Variables on Both Sides

Solving Inequalities with Fractions and Variables on Both Sides

When an inequality contains fractions or variables on both sides, clear fractions by multiplying both sides by the least common denominator (LCD), then solve as usual.

Example: Solving with Fractions

• Solve $\frac{1}{4}(x + 2) \geq \frac{1}{12} - \frac{1}{3}x$

• Multiply both sides by 12 (LCD): $12 \times \frac{1}{4}(x + 2) \geq 12 \times \frac{1}{12} - 12 \times \frac{1}{3}x$

• $3(x + 2) \geq 1 - 4x$

• $3x + 6 \geq 1 - 4x$

• $3x + 4x \geq 1 - 6$

• $7x \geq -5$

• $x \geq -\frac{5}{7}$

• Interval Notation: $\left[ -\frac{5}{7}, \infty \right)$

Example: Variables on Both Sides

• Solve $\frac{1}{3}(x + 1) \geq \frac{2}{3}(3 + 2x)$

• Multiply both sides by 3: $x + 1 \geq 2(3 + 2x)$

• $x + 1 \geq 6 + 4x$

• $x - 4x \geq 6 - 1$

• $-3x \geq 5$

• Divide by -3 (reverse the inequality): $x \leq -\frac{5}{3}$

• Interval Notation: $(-\infty, -\frac{5}{3}]$

Key Points

• Always reverse the inequality when multiplying or dividing by a negative number.

• Express solutions in interval notation and graph them on a number line for clarity.