Solving Linear Inequalities: Concepts, Methods, and Interval Notation

Linear Inequalities

Introduction to Inequalities

Linear inequalities are mathematical statements that compare two expressions using inequality symbols rather than an equals sign. Unlike equations, which have a finite set of solutions, inequalities often have infinitely many solutions represented as intervals on the number line.

• Equation: Uses the equals sign (=) and typically has solutions like $x = a$.

• Inequality: Uses symbols such as <, >, ≤, ≥ and has solution sets that may include infinitely many values.

Key Inequality Symbols:

• < : Less than

• > : Greater than

• ≤ : Less than or equal to

• ≥ : Greater than or equal to

Differences Between Equations and Inequalities

• Equations: Solutions are specific values (e.g., $x = 3$).

• Inequalities: Solutions are ranges or intervals (e.g., $x < 2$ means all $x$ less than 2).

• Solution to an Inequality: Expressed as $x < a$ or $x > a$, and often written in interval notation.

Interval Notation

Interval notation is a concise way to describe sets of numbers that satisfy an inequality.

• Open Interval: $(a, b)$ means all $x$ such that $a < x < b$.

• Closed Interval: $[a, b]$ means all $x$ such that $a ext{≤} x ext{≤} b$.

• Half-Open Intervals: $(a, b]$ or $[a, b)$

• Infinite Intervals: $(- ext{∞}, a)$ or $(b, ext{∞})$

Solving Linear Inequalities

To solve a linear inequality, use similar steps as solving linear equations, but pay special attention to multiplying or dividing by negative numbers, which reverses the inequality sign.

• Step 1: Isolate the variable on one side.

• Step 2: Simplify both sides as needed.

• Step 3: If you multiply or divide both sides by a negative number, reverse the inequality sign.

• Step 4: Express the solution in interval notation and graph it on a number line.

Examples

• Example 1: Solve $3x - 2 > 1$

  • Add 2 to both sides: $3x > 3$

  • Divide by 3: $x > 1$

  • Interval Notation: $(1, ext{∞})$

• Example 2: Solve $2x + 3 < -1$

  • Subtract 3: $2x < -4$

  • Divide by 2: $x < -2$

  • Interval Notation: $(- ext{∞}, -2)$

• Example 3: Solve $3(x - 2) > 4x + 7$

  • Expand: $3x - 6 > 4x + 7$

  • Subtract $3x$: $-6 > x + 7$

  • Subtract 7: $-13 > x$

  • Rewrite: $x < -13$

  • Interval Notation: $(- ext{∞}, -13)$

• Example 4: Solve $4x + 3(x + 1) < -2(x - 3)$

  • Expand: $4x + 3x + 3 < -2x + 6$

  • Combine like terms: $7x + 3 < -2x + 6$

  • Add $2x$: $9x + 3 < 6$

  • Subtract 3: $9x < 3$

  • Divide by 9: $x < rac{1}{3}$

  • Interval Notation: $(- ext{∞}, rac{1}{3})$

Graphing Solutions

Graphing the solution to an inequality involves marking the interval on a number line. Use an open circle for < or > (not including the endpoint), and a closed circle for ≤ or ≥ (including the endpoint).

Summary Table: Inequality Symbols and Their Meanings

Symbol	Meaning	Interval Notation	Graph Representation
<	Less than	$( -\infty, a )$	Open circle at $a$, shade left
>	Greater than	$( a, \infty )$	Open circle at $a$, shade right
≤	Less than or equal to	$( -\infty, a ]$	Closed circle at $a$, shade left
≥	Greater than or equal to	$( a, \infty ]$	Closed circle at $a$, shade right

Important Notes

• When multiplying or dividing both sides of an inequality by a negative number, reverse the inequality sign.

• Always check your solution by substituting values from the interval into the original inequality.

Additional info: Some context and steps were inferred to clarify the solving process and interval notation, as the original notes were fragmented.