Linear Inequalities and Interval Notation: Precalculus Study Notes

Linear Inequalities

Interval Notation

Interval notation is a concise way to express solution sets for inequalities, replacing set notation. It is widely used in precalculus to describe the range of values that satisfy an inequality.

• Closed Interval: Includes both endpoints. Written as $[a, b]$ and represents $a \\leq x \\leq b$.

• Open Interval: Excludes both endpoints. Written as $(a, b)$ and represents $a < x < b$.

• Half-Closed (Half-Open) Interval: Includes one endpoint. Written as $[a, b)$ or $(a, b]$ and represents $a \\leq x < b$ or $a < x \\leq b$.

Infinity Symbols: When solution sets have no upper or lower bound, use $\\infty$ or $-\\infty$ in interval notation. These symbols are always paired with parentheses, never brackets.

• Example: $x \\geq 2$ is written as $[2, \\infty)$.

Graphing Intervals on a Number Line

Intervals can be represented visually on a number line. Closed endpoints are shown with solid dots, open endpoints with hollow dots.

• Example: $[1, 5)$ is a solid dot at 1, hollow dot at 5, and a line connecting them.

Solving Linear Inequalities

Linear inequalities are similar to linear equations but use inequality symbols ($<, \\leq, >, \\geq$) instead of an equals sign. The same algebraic techniques apply, with one important exception: multiplying or dividing both sides by a negative number reverses the inequality symbol.

• Steps to Solve:

  1. Isolate the variable on one side.

  2. Simplify both sides as needed.

  3. If you multiply or divide by a negative, flip the inequality sign.

  4. Express the solution in interval notation and graph it.

• Example: Solve $-2x + 6 < 8$.

  • Subtract 6: $-2x < 2$

  • Divide by -2 (flip sign): $x > -1$

  • Interval notation: $(-1, \\infty)$

Fractions and Variables on Both Sides

Inequalities may include fractions and variables on both sides. The solution process is similar to solving linear equations: clear fractions, collect like terms, and isolate the variable.

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

  • Multiply both sides by 6 to clear denominators: $3(x+1) \\geq 2(2+2x)$

  • Expand: $3x + 3 \\geq 4 + 4x$

  • Subtract $3x$: $3 \\geq 4 + x$

  • Subtract 4: $-1 \\geq x$ or $x \\leq -1$

  • Interval notation: $(-\\infty, -1]$

Summary Table: Interval Notation Types

Type	Set Notation	Interval Notation	Number Line Representation
Closed Interval	$\\{x \\mid a \\leq x \\leq b\\}$	$[a, b]$	Solid dots at $a$ and $b$
Open Interval	$\\{x \\mid a < x < b\\}$	$(a, b)$	Hollow dots at $a$ and $b$
Half-Closed Interval	$\\{x \\mid a \\leq x < b\\}$ or $\\{x \\mid a < x \\leq b\\}$	$[a, b)$ or $(a, b]$	Solid dot at closed endpoint, hollow at open

Practice Problems

• Express $1 \\leq x \\leq 5$ in interval notation: $[1, 5]$

• Express $x < 7$ in interval notation: $(-\\infty, 7)$

• Solve $2 + 13 > 19$ and express in interval notation: $x > 4$, $[4, \\infty)$