Solving Linear Inequalities and Expressing Solution Sets in Interval Notation

Solving Linear Inequalities

Understanding Linear Inequalities

Linear inequalities are mathematical statements involving a linear expression set in relation to another expression using inequality symbols such as <, >, ≤, or ≥. Solving these inequalities involves finding all values of the variable that make the inequality true.

• Key Terms:

  • Inequality: A statement that compares two expressions using <, >, ≤, or ≥.

  • Solution Set: The set of all values that satisfy the inequality.

  • Interval Notation: A way to represent the solution set using intervals on the real number line.

Example Problem

Consider the inequality:

$6 - (x + 5) \geq 6 - 6x$

• Step 1: Simplify both sides

  • Expand the left side: $6 - x - 5$

  • Simplify: $1 - x \geq 6 - 6x$

• Step 2: Collect like terms

  • Add $6x$ to both sides: $1 - x + 6x \geq 6$

  • Simplify: $1 + 5x \geq 6$

• Step 3: Isolate the variable

  • Subtract 1 from both sides: $5x \geq 5$

  • Divide both sides by 5: $x \geq 1$

• Step 4: Express the solution set in interval notation

  • The solution set is $[1, \infty)$

Graphing the Solution Set on a Number Line

• Draw a number line.

• Shade the region starting at $x = 1$ and extending to the right (towards infinity).

• Use a solid dot at $x = 1$ to indicate that 1 is included in the solution set.

Interval Notation

• Closed Interval: $[a, b]$ includes both endpoints $a$ and $b$.

• Open Interval: $(a, b)$ excludes both endpoints.

• Half-Open Interval: $[a, b)$ or $(a, b]$ includes one endpoint.

• Infinity: Always use parentheses with $\infty$ or $-\infty$ since infinity is not a number.

Summary Table: Solution Set Types

Type of Solution	Interval Notation	Graphical Representation
No Solution	$\emptyset$	No shading on the number line
All Real Numbers	$(-\infty, \infty)$	Entire number line shaded
Specific Interval (e.g., $x \geq 1$)	$[1, \infty)$	Shading from $x=1$ to the right

Additional info:

• When solving inequalities, if you multiply or divide both sides by a negative number, you must reverse the inequality sign.

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