A step-by-step guide to inequality transformations, negative divisors, boundaries, graphs, and interval checks.
An inequality describes a set of values
An inequality compares quantities with <, >, ≤, or ≥. Its solution is usually a range rather than one number. The goal is to isolate the variable while preserving the order relationship.
Adding or subtracting the same value on both sides preserves the order. Multiplying or dividing by the same positive value also preserves it. The special rule appears when the multiplier or divisor is negative.
Why the sign reverses with a negative number
On the number line, 5 is greater than 2. Multiplying both by −1 reflects both points across zero: −5 is less than −2. Reflection reverses left and right, so the comparison symbol must reverse to remain true.
The same rule applies in algebra. Dividing −3x > 9 by −3 gives x < −3. Keeping > would describe the opposite half of the number line and would fail a simple test value.
Worked example: −3x + 6 > 15
Subtract 6 from both sides to obtain −3x > 9. Divide both sides by −3. Because the divisor is negative, reverse > to <: x < −3.
Check x = −4, which lies inside the solution set. The original left side is −3(−4) + 6 = 18, and 18 > 15 is true. Check x = 0 outside the set: 6 > 15 is false. These tests support the result.
Open and closed boundaries
Strict symbols < and > exclude the boundary, so a number-line graph uses an open circle. Symbols ≤ and ≥ include equality and use a closed circle. In interval notation, x < −3 becomes (−∞, −3), while x ≤ −3 becomes (−∞, −3].
Infinity is never included as an endpoint because it is not a real number. Parentheses are always used next to ±∞.
Avoid the most common inequality errors
Do not reverse the sign when merely adding or subtracting a negative number; reversal occurs only when multiplying or dividing the entire comparison by a negative value. Also distribute through grouped terms before deciding which coefficient will divide both sides.
For compound inequalities, apply an operation to all three parts. For expressions containing a variable denominator or nonlinear term, domain and sign analysis may be required; those problems are outside a basic linear inequality method.
How to check a linear inequality solution set
Choose a value clearly inside the proposed interval and substitute it into the original inequality. Then choose a value outside. For x < −3, x = −4 should make the statement true and x = 0 should make it false. If −3 is the boundary, test it separately to decide whether equality is included.
These sample checks can expose a reversed sign, but they do not prove the entire interval. The algebraic transformation supplies that proof. A linear inequality calculator with steps should identify the coefficient used for division and state whether it was positive or negative before displaying the final interval.
Practice when to reverse the inequality sign
Compare 4x − 1 ≤ 11 with −4x − 1 ≤ 11. The first gives x ≤ 3 because division is by positive 4. The second gives −4x ≤ 12 and then x ≥ −3 because division is by negative 4. Writing both examples together makes the reversal rule visible.
To learn how to solve linear inequalities step by step, also practise variables on both sides, fractions, identities, and contradictions. Convert each answer to interval notation and a number-line description. Keep nonlinear, compound, rational, and absolute-value inequalities separate unless the selected method explicitly supports their extra domain or case analysis.