Steps to Solve an Inequality: A Comprehensive Guide

Solving an inequality involves a series of steps to find the values that make the inequality statement true. Here is a comprehensive guide on how to solve an inequality:

Step 1: Understand the Inequality

• Carefully read and understand the given inequality statement. Identify the variable, the inequality symbol (e.g., "<," ">", "<=," or ">="), and the constants or expressions on both sides of the inequality.

Step 2: Isolate the Variable

• Your goal is to isolate the variable on one side of the inequality. This may involve adding, subtracting, multiplying, or dividing by constants or expressions on both sides. When performing these operations, ensure that you maintain the direction of the inequality.

Step 3: Simplify

• Simplify the expression on both sides of the inequality by combining like terms. This makes it easier to work with.

Step 4: Graph the Inequality (Optional)

• If you want a visual representation of the solution set, you can graph the inequality on a number line or coordinate plane. Shade the region that represents the values that satisfy the inequality.

Step 5: Determine the Direction of the Inequality

• Based on the inequality symbol, determine the direction in which the solution set lies. If it's "<," the solution is to the left; if it's ">", the solution is to the right; if it's "<=," the solution includes the boundary; if it's ">=" the solution includes the boundary.

Step 6: Express the Solution Set

• Write down the solution set in the form of an interval notation or set notation, depending on the context. If it's an open interval, use parentheses (e.g., $(-∞, 5)$); if it's a closed interval, use square brackets (e.g., $[2, 8]$); if it's a half-open interval, use a combination of parentheses and brackets (e.g., $(3, 7]$).

Step 7: Check the Solution

• Test the values in the solution set by substituting them into the original inequality. Ensure that they make the inequality true.

Step 8: Express the Solution in Plain Language

• If necessary, express the solution in plain language to explain what it means in the context of the problem.

Step 9: Special Considerations

• Be aware of special cases, such as absolute value inequalities or inequalities with variables in the denominator. These may require additional steps or different techniques.

Step 10: Finalize the Solution

• Review your work to make sure you have accurately solved the inequality. Double-check the direction of the inequality and the form of the solution set.

Here's an example to illustrate these steps:

Example: Solve the inequality $2x - 3 < 9$.

Step 1: Understand the Inequality.

• Variable: $x$
• Inequality: "<"
• Constants/Expressions: $2x - 3$ and $9$

Step 2: Isolate the Variable.

• $2x - 3 < 9$
• Add 3 to both sides: $2x < 12$
• Divide by 2 (a positive number): $x < 6$

Step 3: Simplify.

• The expression is already simplified.

Step 4: Graph the Inequality (Optional).

• Represent the solution set as $(-∞, 6)$ on a number line.

Step 5: Determine the Direction of the Inequality.

• Since it's "<," the solution is to the left.

Step 6: Express the Solution Set.

• The solution set is $(-∞, 6)$.

Step 7: Check the Solution.

• Substitute values, such as $x = 4$, into the inequality to ensure it's true: $2(4) - 3 < 9$, which is true.

Step 8: Express the Solution in Plain Language.

• The values of $x$ that make the inequality $2x - 3 < 9$ true are all real numbers less than 6.

Step 9: Special Considerations.

• None in this case.

Step 10: Finalize the Solution.

• Review your work and confirm that you've solved the inequality correctly.

These steps can be used to solve various types of inequalities, and they provide a systematic approach to finding the solution set.