Solving Linear Inequalities and Expressing Solution Sets in Interval Notation

Solving Linear Inequalities

Understanding Linear Inequalities

Linear inequalities are mathematical statements that relate linear expressions using inequality symbols such as <, ≤, >, or ≥. Solving a linear inequality involves finding all values of the variable that make the inequality true.

• Linear inequality: An inequality that can be written in the form ax + b < c, ax + b ≤ c, ax + b > c, or ax + b ≥ c.

• Solution set: The set of all real numbers that satisfy the inequality.

Steps to Solve Linear Inequalities

1. Distribute any factors and simplify both sides of the inequality.

2. Combine like terms if necessary.

3. Isolate the variable on one side of the inequality.

4. Solve for the variable.

5. Express the solution in interval notation.

6. Graph the solution on a number line.

Example Problem

Solve the inequality and express the solution set in interval notation:

$3(8 - 2x) \geq 38 - 4x$

• Step 1: Distribute the 3 on the left side:

$3 \times 8 - 3 \times 2x \geq 38 - 4x$

$24 - 6x \geq 38 - 4x$

• Step 2: Collect all variable terms on one side and constants on the other:

$24 - 6x + 4x \geq 38$

$24 - 2x \geq 38$

• Step 3: Isolate the variable:

$24 - 2x \geq 38$

$-2x \geq 38 - 24$

$-2x \geq 14$

• Step 4: Divide both sides by -2. Important: When dividing both sides of an inequality by a negative number, reverse the direction of the inequality symbol.

$x \leq \frac{14}{-2}$

$x \leq -7$

• Step 5: Express the solution in interval notation:

The solution set is $(-\infty, -7]$.

Interval Notation

• Interval notation is a way of writing subsets of the real number line. For example, $(-\infty, -7]$ means all real numbers less than or equal to -7.

• Parentheses ( ) indicate that an endpoint is not included; brackets [ ] indicate that an endpoint is included.

Graphing the Solution Set on a Number Line

• Draw a number line and shade all values to the left of -7.

• Place a solid dot at -7 to indicate that -7 is included in the solution set.

Summary Table: Inequality Symbols and Interval Notation

Inequality	Interval Notation	Graphing on Number Line
$x < a$	$(-\infty, a)$	Open circle at $a$, shade left
$x \leq a$	$(-\infty, a]$	Closed circle at $a$, shade left
$x > a$	$(a, \infty)$	Open circle at $a$, shade right
$x \geq a$	$[a, \infty)$	Closed circle at $a$, shade right

Key Points

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

• Express the solution set in interval notation for clarity and precision.

• Graphing helps visualize the solution set on the real number line.

Example Solution

• Given: $3(8 - 2x) \geq 38 - 4x$

• Solution in interval notation: $(-\infty, -7]$