A linear inequality in one variable is solved by the same operations that solve a linear equation — with one additional rule. Dividing or multiplying by a negative number flips the direction of the comparison. That single rule is the only point of divergence, and once it is internalized, every first-degree inequality reduces to a statement of the form x<c or x>c, whose solution is a ray on the number line. The boundary point is always the solution of the corresponding equation, and the inequality determines which side of it to shade.
Definition and Standard Form
A linear inequality in one variable is any inequality that can be written as
ax+b<0
(or with >, ≤, or ≥ in place of <), where a and b are real constants and a=0. The variable x appears only to the first power — no squares, no roots, no x in a denominator.
The solution is always a ray: a half-line beginning at one point and extending to infinity in one direction. The boundary point is x=−ab, the solution of the corresponding linear equationax+b=0. The inequality symbol determines which side of this boundary the solution occupies and whether the boundary itself is included.
Not every linear inequality arrives in standard form. The inequality 3x−7>2x+5 is linear because, after collecting terms, it reduces to x−12>0. Recognition is the same as for equations: if the variable appears only to the first power with no compositions, the inequality is linear regardless of its initial presentation.
Solving with the Properties of Inequality
Two properties govern the manipulation of linear inequalities, directly paralleling the properties used for equations.
The addition property states that adding or subtracting the same quantity on both sides preserves the direction of the inequality. If x−5>3, then adding 5 to both sides gives x>8. The direction > is unchanged.
The multiplication property has two cases. Multiplying or dividing by a positive number preserves the direction: from 3x≤21, dividing by 3 gives x≤7. Multiplying or dividing by a negative number reverses the direction: from −2x≤8, dividing by −2 gives x≥−4. The inequality flips from ≤ to ≥.
The reversal is the entire difference between solving equations and solving inequalities. Every other step — collecting terms, distributing, clearing fractions — is mechanically identical. Missing the flip when dividing by a negative is the single most common error, and it produces the exact complement of the correct solution: the wrong half of the number line.
Worked Examples
A single-step inequality requires one operation. The inequality 3x>12 is solved by dividing both sides by 3 (positive, no flip): x>4. The solution in interval notation is (4,∞).
A multi-step inequality demands the same sequence as a multi-step equation. Consider 4x−7≤2x+11. Subtract 2x from both sides: 2x−7≤11. Add 7: 2x≤18. Divide by 2: x≤9. The solution is (−∞,9].
An inequality requiring a direction flip: −2x+5>13. Subtract 5: −2x>8. Divide by −2 and reverse the inequality: x<−4. The solution is (−∞,−4).
A more involved example: 5−3(2x+1)≥4x−8. Distribute: 5−6x−3≥4x−8. Simplify the left: 2−6x≥4x−8. Add 6x: 2≥10x−8. Add 8: 10≥10x. Divide by 10: 1≥x, or equivalently x≤1. The solution is (−∞,1].
Graphing the Solution
The solution to a linear inequality is represented on the number line by marking the boundary point and shading the appropriate direction.
For x>4: place an open dot at 4 (excluded because the inequality is strict) and shade to the right. The interval is (4,∞).
For x≤9: place a solid dot at 9 (included because the inequality is non-strict) and shade to the left. The interval is (−∞,9].
The boundary point is always the solution of the corresponding equation. The inequality 4x−7≤2x+11 has the boundary at x=9, the solution of 4x−7=2x+11. The inequality symbol tells which side to shade, and whether the dot is solid or open.
This connection is not a coincidence. The equation ax+b=0 identifies the single value where the two sides of the inequality are equal. On one side of that value, ax+b is positive; on the other, it is negative. The inequality selects the appropriate side.
Compound Linear Inequalities
A compound inequality joins two linear inequalities into a single condition. The conjunction (AND) and disjunction (OR) produce fundamentally different solution sets.
A three-part chain like −3<2x+1≤7 is a conjunction: both −3<2x+1 and 2x+1≤7 must hold. Operations are applied to all three parts simultaneously. Subtract 1: −4<2x≤6. Divide by 2: −2<x≤3. The solution is the half-open interval (−2,3].
A disjunction like x+3<−1 or x+3>5 requires solving each part independently. From the first: x<−4. From the second: x>2. The solution is the union (−∞,−4)∪(2,∞) — two separate rays with a gap between them.
The choice between AND and OR depends on the problem's structure. Conjunctions narrow the solution set by requiring both conditions; the result is the intersection of two rays, which is either a bounded interval or empty. Disjunctions broaden it by accepting either condition; the result is the union of two rays, which is either two separate rays or the entire line.
Inequalities with Fractions and Decimals
Fractional coefficients are handled exactly as in linear equations: multiply every term on both sides by the least common denominator to clear all fractions. The critical requirement is that the LCD must be positive, which it always is when the denominators are numerical constants.
Consider 3x−4x+2>1. The LCD of 3 and 4 is 12. Multiplying every term by 12:
4x−3(x+2)>12
Distributing and collecting:
4x−3x−6>12
x−6>12
x>18
The solution is (18,∞). The multiplication by 12 is safe because 12>0, so the direction is preserved.
Decimal coefficients are cleared in the same way, by multiplying by the appropriate power of 10. The inequality 0.4x−1.5≤0.1x+3 becomes 4x−15≤x+30 after multiplying by 10, which simplifies to 3x≤45 and then x≤15.
If the variable appeared in a denominator, the inequality would no longer be linear — it would be a rational inequality, requiring sign analysis rather than simple clearing.
Special Cases
When the variable terms cancel during simplification, the inequality reduces to a comparison between two constants. The result is either universally true or universally false.
An always-true inequality arises when the remaining statement is valid. The inequality 2(x+4)>2x+3 distributes to 2x+8>2x+3. Subtracting 2x gives 8>3, which holds regardless of x. Every real number satisfies the original inequality, and the solution set is (−∞,∞).
A never-true inequality arises when the remaining statement is false. The inequality 5(x−1)≤5x−9 distributes to 5x−5≤5x−9. Subtracting 5x gives −5≤−9, which fails for every x. The solution set is empty — no real number satisfies the inequality.
These cases parallel the identity and contradiction outcomes in linear equations. The mechanism is the same: the variable disappears, and the truth or falsity of the resulting constant statement determines whether the solution set is everything or nothing.
Literal Inequalities
A literal inequality contains multiple variables, and the task is to isolate one variable in terms of the others. The procedure follows the same rules as numerical inequalities, but with an added complication: when dividing by a variable, its sign may not be known.
Consider solving ax+b>c for x. Subtracting b gives ax>c−b. To divide by a, the sign of a matters. If a>0, the direction is preserved: x>ac−b. If a<0, the direction reverses: x<ac−b. Both cases must be stated, because without knowing a, neither can be discarded.
This case-splitting does not arise in numerical inequalities, where the coefficient's sign is always visible. In literal inequalities, the answer is often conditional: the solution takes one form when a parameter is positive and another when it is negative. Stating both cases explicitly is not optional — omitting one produces an answer that is wrong half the time.
The same caution applies to multiplying by a variable expression whose sign is unknown. This is why, in more complex inequalities, sign analysis replaces direct manipulation: it avoids the need to know the sign of any expression in advance.