Solving Linear Inequalities and Expressing Solution Sets in Interval Notation

Linear Inequalities

Solving Linear Inequalities

Linear inequalities are mathematical statements involving a linear expression set in relation to another expression using inequality symbols such as <, >, ≤, or ≥. The solution to a linear inequality is the set of all real numbers that make the inequality true.

• Definition: A linear inequality is an inequality that involves a linear function. For example, $ax + b \geq c$.

• Key Steps to Solve:

  1. Distribute and simplify both sides of the inequality.

  2. Collect like terms and isolate the variable on one side.

  3. Solve for the variable.

  4. Express the solution set using interval notation.

  5. Graph the solution set on a number line.

Example: Solving the Given Inequality

Consider the inequality:

$3(x - 1) - 5(x + 1) \geq -2x - 2$

• Step 1: Distribute

  • $3(x - 1) = 3x - 3$

  • $-5(x + 1) = -5x - 5$

  So, the inequality becomes:

  $3x - 3 - 5x - 5 \geq -2x - 2$

• Step 2: Combine Like Terms

  • Left side: $3x - 5x = -2x$

  • $-3 - 5 = -8$

  So, $-2x - 8 \geq -2x - 2$

• Step 3: Isolate the Variable

  • Add $2x$ to both sides:

  • $-2x - 8 + 2x \geq -2x - 2 + 2x$

  • $-8 \geq -2$

• Step 4: Analyze the Result

  • The result $-8 \geq -2$ is always true for any real value of $x$.

  • This means the solution set is all real numbers.

• Step 5: Express in Interval Notation

  • All real numbers: $(-\infty, \infty)$

• Step 6: Graph on a Number Line

  • Shade the entire number line to represent all real numbers.

Interval Notation

Interval notation is a way of writing subsets of the real number line. For all real numbers, the interval notation is $(-\infty, \infty)$.

• Examples:

  • $x \geq 2$ is $[2, \infty)$

  • $x < 5$ is $(-\infty, 5)$

  • All real numbers is $(-\infty, \infty)$

Graphing Solution Sets on a Number Line

To graph the solution set $(-\infty, \infty)$ on a number line:

• Draw a horizontal line representing the real numbers.

• Shade or mark the entire line to indicate all real numbers are included.

Summary Table: Solution Set Types

Inequality	Interval Notation	Number Line Representation
$x \geq a$	$[a, \infty)$	Shade right from $a$ (including $a$)
$x < b$	$(-\infty, b)$	Shade left from $b$ (not including $b$)
All real numbers	$(-\infty, \infty)$	Shade entire line
No solution	$\emptyset$	No shading

Additional info:

• When solving inequalities, if the variable cancels and the resulting statement is always true (like $-8 \geq -2$), the solution is all real numbers.

• If the resulting statement is always false (like $-8 \geq 2$), the solution is the empty set ($\emptyset$).