Solving and Graphing Linear Inequalities

Equations, Inequalities, and Applications

Solving Linear Inequalities

Linear inequalities are mathematical statements that compare two expressions using inequality signs such as <, >, ≤, or ≥. The solution to an inequality is the set of all values that make the statement true. Solving inequalities involves similar steps to solving equations, with special attention to operations involving negative numbers.

• Definition: An inequality is a statement that two expressions are not necessarily equal, but one may be greater or less than the other.

• Key Inequality Signs: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to)

• Solution Set: All values of the variable that make the inequality true.

Steps for Solving Inequalities

• Simplify both sides: Remove parentheses and combine like terms.

• Isolate the variable: Use addition, subtraction, multiplication, or division to solve for the variable.

• Reverse the inequality sign: If you multiply or divide both sides by a negative number, reverse the direction of the inequality sign.

Example:

• Original Inequality: $4x < -4$

• Divide both sides by 4: $x < -1$

Operations with Positive and Negative Numbers

• Adding/Subtracting a Positive Number: The direction of the inequality does not change.

• Multiplying/Dividing by a Positive Number: The direction of the inequality does not change.

• Multiplying/Dividing by a Negative Number: The direction of the inequality reverses.

Example:

• Original Inequality: $-2x \geq 6$

• Divide both sides by -2 (reverse the sign): $x \leq -3$

Graphing Solutions to Inequalities

After solving an inequality, the solution can be represented on a number line. Use an open circle for < or > and a closed circle for ≤ or ≥. Shade the region that represents all possible solutions.

• Example: $x < -1$ is graphed with an open circle at -1 and shading to the left.

• Example: $x \leq -3$ is graphed with a closed circle at -3 and shading to the left.

Helpful Hints

• Always combine like terms and remove parentheses before solving for the variable.

• Remember to reverse the inequality sign when multiplying or dividing by a negative number.

Sample Problems

• Solve and graph: $4x - 6 > x - 5$

• Solve and graph: $7 - 3x \leq 19$

Summary Table: Effects of Operations on Inequalities

Operation	Effect on Inequality Sign
Add/Subtract Positive Number	No change
Multiply/Divide by Positive Number	No change
Multiply/Divide by Negative Number	Reverse direction

Example: If $a < b$, then $-a > -b$ after multiplying both sides by -1.

Additional info: Solving and graphing inequalities is a foundational skill for algebra, used in applications such as determining ranges for variables in real-world problems.