Linear Inequalities in One Variable

An inequality is a statement that two quantities are not necessarily equal — one is less than, greater than, or comparable to the other. A linear inequality in one variable is any statement of the form $ax + b < 0$, $ax + b \le 0$, $ax + b > 0$ or $ax + b \ge 0$, where $a \ne 0$. The strict symbols $<$ and $>$ exclude equality; the weak symbols $\le$ and $\ge$ allow it. Real life is full of them — a lift rated for "at most $8$ persons", a phone plan costing "under ₹$500$", a pass mark of "at least $40\%$" are all linear inequalities in disguise.

Solving an inequality means finding every value of the variable that makes the statement true. Unlike an equation, which usually pins the variable to one or two numbers, an inequality almost always has a whole range of solutions — an interval on the number line.

Most of the rules you already use for equations carry over unchanged. You may add or subtract the same number from both sides, and you may multiply or divide both sides by the same positive number. The one rule that catches students out is the sign-flip:

For example, $-3x \ge 6 \Rightarrow x \le -2$ — dividing by $-3$ turns $\ge$ into $\le$. The table below summarises which operations preserve the direction of the inequality and which reverse it:

The standard algebraic method mirrors solving an equation: clear fractions by multiplying through by the (positive) LCM of the denominators, expand any brackets, gather the variable terms on one side and the constants on the other, then divide by the coefficient of $x$ — remembering to flip the symbol only if that coefficient is negative. The goal is always to isolate $x$ on one side as $x < k$, $x \le k$, $x > k$ or $x \ge k$.

The solution is then written either in set-builder form, such as $\{x : x \le -2,\ x \in \mathbb{R}\}$, or as an interval, such as $(-\infty, -2]$. Recall the bracket convention: a square bracket includes the endpoint (it matches $\le$ or $\ge$), a round bracket excludes it (matching $<$ or $>$), and $\pm\infty$ always takes a round bracket because it is not a number.

On the number line we use an open circle ($\circ$) at the endpoint for a strict inequality and a closed circle ($\bullet$) for a weak one, then shade the ray of all values that satisfy it. The picture and the interval carry exactly the same information — the open/closed circle on the line is the visual twin of the round/square bracket in the interval.

When the variable is restricted to the integers ($x \in \mathbb{Z}$) or naturals ($x \in \mathbb{N}$) rather than all reals, the same algebra applies but the final answer is a discrete list, for instance $\{-2, -1, 0, 1\}$, not a continuous interval. Always read the domain stated in the question before writing the answer.

Word problems are where inequalities earn their keep. Translate the English into algebra phrase by phrase: "at least" and "minimum" become $\ge$, "at most" and "maximum" become $\le$, "more than" and "exceeds" become $>$, "less than" and "fewer than" become $<$. Define the variable clearly, build the inequality, solve it, and then sanity-check the answer against the real-world context — a number of buses or students must be a whole number, so you often round to the nearest sensible integer at the end.

Deeper Insight — why the sign-flip is unavoidable: The reversal rule is not an arbitrary convention you must memorise; it is forced by what "less than" actually means on the number line. Consider the plain truth $2 < 5$. Now multiply both sides by $-1$: the numbers $-2$ and $-5$ sit on the negative side, where the order is mirrored — $-5$ lies further left than $-2$, so $-2 > -5$. Multiplying by any negative number reflects every point across zero, and reflection turns "to the left of" into "to the right of", which is exactly what flipping the symbol records. This is also why you should never multiply an inequality by an expression like $x$ whose sign you do not know — you would not know whether to flip. Keep the operations transparent: isolate the variable using additions and positive multipliers wherever possible, and reach for a negative multiplier only when you consciously remember to reverse the symbol.