Inequalities Step by Step: Solving, Graphing, and Checking Compound Answers

Overview

An inequality compares two expressions using a symbol such as <, >, ≤, or ≥. For example, x + 3 > 7 asks for every value of x that makes the statement true. Solving gives x > 4, so the answer is an entire set of numbers, not only x = 4.

The basic process resembles solving a linear equation:

1. Simplify each side when needed.
2. Use inverse operations to isolate the variable.
3. Reverse the inequality sign if you multiply or divide by a negative number.
4. Write the answer in a useful form, such as inequality, set-builder, interval, or number-line notation.
5. Test a value from the proposed solution to confirm the result.

The sign-reversal rule is the detail students most often miss. Adding or subtracting the same number from both sides does not change the direction. Multiplying or dividing both sides by a positive number also leaves it unchanged. Multiplying or dividing by a negative number reverses it: -2x > 8 becomes x < -4.

Checklist by scenario

One-step inequalities

Undo the operation attached to the variable. For x - 5 ≤ 9, add 5 to both sides:

x - 5 + 5 ≤ 9 + 5
x ≤ 14

For a negative coefficient, remember to reverse the sign. In -3x ≥ 12, divide by -3:

x ≤ -4

To graph x ≤ -4, place a closed dot at -4 and shade to the left. A closed dot includes the endpoint because the symbol is ≤.

Multi-step linear inequalities

First distribute, combine like terms, and move variable terms to one side. Consider:

3(x - 2) + 4 > 2x + 9

1. Distribute: 3x - 6 + 4 > 2x + 9.
2. Combine constants: 3x - 2 > 2x + 9.
3. Subtract 2x: x - 2 > 9.
4. Add 2: x > 11.

The graph has an open dot at 11 and shading to the right. Use an open dot for < or >, because the endpoint is excluded.

If the variable disappears, examine the remaining statement. For example, simplifying an inequality might produce 5 < 9, which is always true; the solution is all real numbers. If it produces 7 ≥ 12, which is false, there is no solution.

Compound inequalities with “and”

An “and” compound inequality requires both conditions to be true. A chained inequality such as

2 < x + 3 ≤ 8

can be solved by subtracting 3 from all three parts:

-1 < x ≤ 5

This means the answer lies between -1 and 5, excluding -1 and including 5. In interval notation, write (-1, 5]. The parenthesis at -1 means “not included,” while the bracket at 5 means “included.” On a number line, use an open dot at -1, a closed dot at 5, and shade between them.

Compound inequalities with “or”

“Or” means that either condition can be true. For example:

x < -2 or x ≥ 4

Graph two separate rays: an open dot at -2 shaded left, and a closed dot at 4 shaded right. In interval notation, the answer is (-∞, -2) ∪ [4, ∞). The union symbol, ∪, joins the two possible regions.

When solving a compound statement written as two separate inequalities, solve each part independently. Then ask whether the original wording requires an intersection, meaning values shared by both conditions, or a union, meaning values meeting either condition.

Absolute-value inequalities

Absolute value measures distance from zero, so it naturally creates one-sided or two-sided ranges.

For a statement of the form |x| < a, where a is positive, write a compound inequality:

-a < x < a

Thus, |x - 2| ≤ 5 becomes -5 ≤ x - 2 ≤ 5. Adding 2 throughout gives -3 ≤ x ≤ 7.

For a statement of the form |x| > a, split it into two cases:

x < -a or x > a

So |x + 1| > 3 becomes x + 1 < -3 or x + 1 > 3, giving x < -4 or x > 2. The graph has two outside rays, not one interval between the endpoints.

What to double-check

Before accepting an answer, use this verification checklist:

• Check the original inequality. Substitute a convenient value from the shaded region, not merely the final rearranged line.
• Check an excluded endpoint. If the answer is x > 3, test 3. It should make the original statement false or fail the strict comparison.
• Review negative operations. Circle every step where you divide or multiply by a negative number and confirm that the sign changed.
• Match the graph to the notation. Parentheses correspond to open endpoints; brackets correspond to closed endpoints. Infinity always uses parentheses.
• Check the direction of shading. Values greater than a number extend right; values less than a number extend left.
• Test both regions for “or.” A split answer should have one valid test value in each shaded section.

A quick numerical check is often enough. If you claim x > 11, test 12 in the original inequality and then test a value such as 10. The first should work and the second should not. This check is especially useful when distributing negatives or simplifying several terms.

Common mistakes

Forgetting to reverse the sign

This is the defining difference between many inequality problems and ordinary equation problems. If you divide by -4, reverse < to >, or ≥ to ≤. Write a short note beside the step if necessary.

Using the wrong endpoint

An open circle represents an endpoint that is not included: < or >. A closed circle represents an included endpoint: ≤ or ≥. The graph must agree with the algebraic symbol.

Confusing “and” with “or”

An “and” answer usually creates an overlap or a single interval. An “or” answer may create two separate rays or intervals. Read the words before choosing whether to use an intersection or a union.

Checking only the final line

Substituting into a rearranged inequality can hide an earlier sign error. Use the original problem for the final check, especially after distributing or combining fractions.

Stopping before presenting the solution set

A number-line graph, interval notation, or set-builder notation can make the answer clearer. For example, x ≥ 2, [2, ∞), and {x | x ≥ 2} describe the same solution set in different forms.

When to revisit

Return to this checklist whenever a new problem introduces a negative coefficient, a compound statement, an absolute-value expression, or a request for a graph. It is also useful before a quiz or test: solve a few practice problems, then review only the step where your work tends to fail.

For ongoing study, keep a short error log with four columns: original problem, step where the error occurred, corrected rule, and a similar problem to retry. A practice planner can help you track whether your mistakes involve sign reversal, endpoint notation, distribution, or choosing between “and” and “or.” If the algebra includes quadratics, review quadratic equations step by step; for basic rearranging, compare your process with this guide to solving linear equations.

Finally, use this five-question routine before submitting any inequality answer:

1. Did I isolate the variable correctly?
2. Did I reverse the sign after multiplying or dividing by a negative?
3. Does the endpoint symbol match the inequality?
4. Does the number-line shading match the answer?
5. Does a test value satisfy the original inequality?

If all five checks pass, your solution is not only simplified—it is supported by a clear and reusable verification process.