Solving Linear Inequalities and Expressing Solution Sets

Solving Linear Inequalities

Understanding Linear Inequalities

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

• Linear inequality: An inequality that involves a linear expression, such as $ax + b \geq c$.

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

• Interval notation: A concise way to describe sets of numbers on the real number line.

Example Problem: Solving and Graphing a Linear Inequality

Consider the inequality:

$4(8 - 2x) \geq 28 - 6x$

• Step 1: Distribute and simplify both sides.

$4 \times 8 - 4 \times 2x \geq 28 - 6x$

$32 - 8x \geq 28 - 6x$

• Step 2: Collect like terms.

Add $8x$ to both sides:

$32 \geq 28 + 2x$

Subtract $28$ from both sides:

$4 \geq 2x$

• Step 3: Solve for $x$.

Divide both sides by $2$:

$2 \geq x$

Or, equivalently:

$x \leq 2$

• Step 4: Express the solution set in interval notation.

The solution set is all real numbers less than or equal to $2$:

Interval notation: $(-\infty, 2]$

Graphing the Solution Set on a Number Line

To graph $x \leq 2$ on a number line:

• Draw a solid dot or bracket at $2$to indicate that $2$ is included (since the inequality is "less than or equal to").

• Shade the number line to the left of $2$to represent all numbers less than $2$.

Table: Solution Set Representation

Form	Representation
Inequality	$x \leq 2$
Interval Notation	$(-\infty, 2]$
Number Line	Solid dot or bracket at $2$, shaded to the left

Key Points

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

• Interval notation uses parentheses for open intervals (not including the endpoint) and brackets for closed intervals (including the endpoint).

• No solution is denoted by the empty set symbol $\varnothing$ or $\emptyset$.

Example

Problem: Solve $3x - 5 > 7$

• Add $5$to both sides: $3x > 12$

• Divide by $3$:$x > 4$

• Interval notation: $(4, \infty)$

Additional info: The process of solving linear inequalities is foundational for understanding more advanced topics in algebra and precalculus, such as systems of inequalities and absolute value inequalities.