Solving Linear Inequalities
An inequality compares values rather than asserting exact equality. Many equation-solving techniques still apply: add or subtract the same quantity on both sides, simplify, and isolate the variable. The critical difference is that multiplying or dividing both sides by a negative quantity reverses the inequality direction.
Learning objectives
- Interpret <, >, ≤ and ≥
- Solve one-step and multi-step inequalities
- Apply the negative multiplication/division reversal rule
- Write solution sets compactly
- Check candidate values against an inequality
Source scope
Lesson 9, pp. 67-70
The article paraphrases and restructures the supplied source. Source-branded names, personal names, promotional material and original test questions are not reproduced.
Core concepts and decision rules
Inequality symbols describe order
x > 4 means every allowed x is greater than 4; x ≥ 4 also includes 4 itself. The equality bar changes endpoint inclusion.
Addition and subtraction preserve order
Adding or subtracting the same quantity on both sides does not reverse the inequality.
Negative scaling reverses order
If a < b, then -a > -b. Therefore an inequality symbol must reverse when both sides are multiplied or divided by a negative quantity.
A solution is usually a set
Unlike a basic linear equation with one solution, a linear inequality commonly represents infinitely many values.
Checking uses representative values
Test a value from the proposed solution region and, when useful, a value outside it. This checks both algebra and inequality direction.
Step-by-step method
Worked examples
Problem: Solve 3x + 4 < 19.
- Subtract 4: 3x < 15.
- Divide by positive 3; the symbol stays the same.
Problem: Solve -4y + 2 ≥ 18.
- Subtract 2: -4y ≥ 16.
- Divide by -4 and reverse ≥ to ≤.
Problem: Solve 5p - 3 > 2p + 9.
- Subtract 2p: 3p - 3 > 9.
- Add 3: 3p > 12.
- Divide by 3.
How to reason through solving linear inequalities
1. Identify the mathematical structure
Before calculating, classify what you are looking at. Decide whether the expression is a sum, product, quotient, power, equation, inequality, graph or system. Then identify the terms, signs, grouping symbols and variables that control the next legal move. This classification step prevents a common failure mode in algebra: applying a familiar rule to the wrong structure.
Use notation as information. A sign attached to a term belongs to that term; parentheses define a unit of work; an exponent applies to its stated base; and an equals or inequality symbol separates two related expressions. Read the structure before manipulating it.
2. Preserve equivalence or implication
Algebra is not a sequence of arbitrary rearrangements. Each line should follow from the previous line by a named rule. When simplifying an expression, preserve its value for every admissible input. When solving an equation, preserve the equality unless you knowingly use an operation such as squaring that can introduce extra candidates and therefore requires a final check.
A useful discipline is to ask: What operation did I apply, and to what complete object did I apply it? This question catches incomplete distribution, partial denominator clearing, lost signs and unbalanced equation operations.
3. Separate exact work from approximation
Keep fractions, powers and radicals exact while the algebra is still being transformed. Approximate decimals are best introduced only when a problem requires a numerical result to a stated precision. Exact intermediate forms are easier to verify and avoid cumulative rounding drift.
When an application does require rounding, retain enough guard digits during the calculation and round only the reported result. This is an illustrative good-calculation practice rather than a numerical requirement from the source.
4. Build an independent check
Use a check that is different from the step that produced the answer. Substitute a solved variable into the original equation, expand proposed factors, square a simplified radical, test a point on a graph, or evaluate both original and simplified expressions at a convenient value. Independent checks are more valuable than rereading the same arithmetic because they test the relationship from another direction.
If the check fails, work backwards through the written transformations until the first inconsistent line appears. Correct that line, not merely the final number.
Quick-reference table
| Rule or idea | How to use it |
|---|---|
| Inequality symbols describe order | x > 4 means every allowed x is greater than 4; x ≥ 4 also includes 4 itself. The equality bar changes endpoint inclusion. |
| Addition and subtraction preserve order | Adding or subtracting the same quantity on both sides does not reverse the inequality. |
| Negative scaling reverses order | If a < b, then -a > -b. Therefore an inequality symbol must reverse when both sides are multiplied or divided by a negative quantity. |
| A solution is usually a set | Unlike a basic linear equation with one solution, a linear inequality commonly represents infinitely many values. |
Common mistakes and controls
- Forgetting to reverse the symbol after dividing by a negative number
- Reversing the symbol after ordinary addition or subtraction
- Treating ≥ as if the endpoint were excluded
- Reporting a single test value instead of a solution set
- Checking only the arithmetic and not the comparison direction
Applications
Constraint modelling
Inequalities are natural for limits: capacity must be at least a target, a dimension must not exceed a maximum, or a resource must remain below a bound.
Classification: Illustrative application unless directly stated as a source concept.
Boundary awareness
The difference between < and ≤ can represent whether a boundary value is prohibited or permitted; preserve that distinction throughout the algebra.
Classification: Illustrative application unless directly stated as a source concept.
Practice and self-check
These questions are newly written for this KEVOS article; they are not copied from the supplied source.
Show answer
Show answer
Show answer
Show answer
Show answer
Show answer
Related KEVOS knowledge
Source fidelity note: Topic selection and instructional sequence are grounded in the supplied algebra source. Mathematical explanations have been paraphrased and reorganised into a web-handbook format. No external standards, company-specific requirements or numerical engineering limits are asserted.