What Are the Rules for Solving Linear Inequalities?
You solve a linear inequality almost exactly like a linear equation, isolate the variable, with one rule that has no equation equivalent.
Add or subtract anything, both sides, the sign stays. $x + 5 \leq 9 \implies x \leq 4$.
Multiply or divide by a positive number, the sign stays. $3x < 12 \implies x < 4$.
Multiply or divide by a negative number, the sign flips. This is the rule. $-2x < 6 \implies x > -3$ (the $<$ became $>$).
The flip is not arbitrary. Multiplying by a negative reverses order on the number line: $2 < 3$ is true, but multiply both sides by $-1$ and $-2 < -3$ is false, you have to write $-2 > -3$ to keep it true. So whenever a negative factor crosses the inequality, the symbol reverses.
Why does the inequality sign flip when you divide by a negative?
Because "bigger" and "smaller" swap places under negation. Picture $4$ and $7$ on the number line, where $4 < 7$. Their negatives $-4$ and $-7$ sit on the mirror-image side, and now $-4 > -7$ (the larger original became the smaller negative).
Dividing or multiplying an inequality by a negative does exactly this mirroring, so the symbol must turn around to stay true. Forgetting this single flip is the number-one error in the topic.
Types of Linear Inequalities
One variable: $3x - 1 > 5$. The solution is an interval on the number line.
Two variables: $2x + y \leq 6$. The solution is a half-plane on the coordinate grid, every point on one side of the boundary line.
A single linear inequality is not the same as a compound inequality, which joins two inequalities with "and"/"or" (like $1 < x \leq 5$). This article stays on the single inequality; the compound case has its own logic and its own page.
Examples of Linear Inequalities
Example 1
Solve $x + 7 < 12$.
Subtract $7$ from both sides:
$x + 7 - 7 < 12 - 7$
$x < 5$.
Final answer: $x < 5$, every value below $5$.
Example 2
Solve $-3x \geq 9$.
A student divides both sides by $-3$ and writes $x \geq -3$.
Here is the wrong path, and it is the most instructive one in this topic. Dividing by $-3$ as if it were a positive leaves the $\geq$ untouched.
Test it. If $x \geq -3$ were right, then $x = 0$ should work: $-3(0) = 0 \geq 9$? That is false. The "solution" includes values that break the original inequality, so it cannot be right.
The rescue: dividing by a negative flips the sign.
$-3x \geq 9$
$x \leq -3$ (divide by $-3$, flip $\geq$ to $\leq$).
Test $x = -4$: $-3(-4) = 12 \geq 9$. ✓
Final answer: $x \leq -3$.
Example 3
Solve $2x + 3 \leq 11$ and graph it.
Subtract $3$:
$2x \leq 8$.
Divide by $2$ (positive, sign stays):
$x \leq 4$.
On the number line: a closed circle at $4$ (because $\leq$ includes $4$), shaded left.
Final answer: $x \leq 4$.
Example 4
Solve $5 - 2x > 1$.
Subtract $5$:
$-2x > -4$.
Divide by $-2$ (negative, flip $>$ to $<$):
$x < 2$.
Final answer: $x < 2$. In interval notation that is $(-\infty, 2)$.
Example 5
Solve $\dfrac{x}{3} - 2 \geq 4$.
Add $2$:
$\dfrac{x}{3} \geq 6$.
Multiply by $3$ (positive, sign stays):
$x \geq 18$.
Final answer: $x \geq 18$, or $[18, \infty)$ in interval notation.
Example 6
Solve the two-variable inequality $x + y \leq 4$ and describe its graph.
First treat the boundary as an equation: $x + y = 4$ is the line through $(4, 0)$ and $(0, 4)$.
Because the relation is $\leq$ (includes equality), the boundary line is solid.
Test a point not on the line, the origin $(0,0)$: $0 + 0 = 0 \leq 4$? True.
So shade the side of the line containing the origin.
Final answer: the solution is the closed half-plane on the origin's side of the solid line $x + y = 4$.
Why Inequalities Run the Real World
"Why not just use equations?" Because most decisions are about staying within a limit, not hitting it exactly, and a limit is an inequality.
Constraints are inequalities by nature. A budget says spending $\leq$ income. A speed limit says $v \leq 60$. A safety factor says load capacity $\geq$ expected load. None of these is an equals sign.
Optimisation lives on inequalities. Linear programming, the method that schedules airlines, routes deliveries, and plans factory output, is built entirely on systems of linear inequalities, finding the best point inside a region they fence off.
The boundary is where the danger is. Treating $\geq$ as $=$ ignores the safety margin. The 1940 Tacoma Narrows Bridge collapse is the textbook reminder that a structure rated for "up to" a load behaves very differently once the inequality is violated, the margin is the point, not the rounding error.
The destination, then, is not "solve for $x$." It is: describe the entire safe region, and know which boundary you must not cross.
Where Linear Inequalities Trip Students Up
Mistake 1: Forgetting to flip the sign with a negative
Where it slips in: any step that divides or multiplies by a negative number.
Don't do this: treat $-2x > 6$ like an equation and write $x > -3$. The memorizer who learned "do the same to both sides" applies it without the negative-number exception.
The correct way: the moment a negative factor crosses the inequality, reverse the symbol: $-2x > 6 \implies x < -3$. A reliable habit is to test one value from your answer back in the original, it catches a missed flip instantly.
Mistake 2: Using the wrong circle when graphing
Where it slips in: drawing the solution on a number line.
Don't do this: the rusher uses a filled dot for $<$ or an open dot for $\leq$ without thinking about which boundary is included.
The correct way: open circle for strict $<$ or $>$ (boundary excluded); closed circle for $\leq$ or $\geq$ (boundary included). The symbol's "or equal to" half is exactly what fills the dot in.
Mistake 3: Shading the wrong half-plane in two variables
Where it slips in: graphing two-variable inequalities.
Don't do this: the second-guesser draws the boundary line correctly, then shades whichever side "looks right."
The correct way: pick a test point not on the line, the origin $(0,0)$ is easiest when the line misses it, substitute it into the inequality, and shade the side it lands on if the statement is true. Never guess the side; test it.
Conclusion
A linear inequality compares two first-degree expressions with $<$, $>$, $\leq$, or $\geq$, and its solution is a range of values.
You solve it like an equation, with one exception: multiplying or dividing by a negative flips the sign.
On a number line, use an open circle for $<$/$>$ and a closed circle for $\leq$/$\geq$.
A two-variable linear inequality graphs as a half-plane, with the correct side found by a test point.
A single linear inequality differs from a compound inequality, which joins two with "and" or "or."
A Practical Next Step
Practice these to solidify your understanding:
Solve $4 - x \leq 7$ and graph it.
Solve $-5x + 2 > 17$ and state the answer in interval notation.
Graph $2x - y > 2$ in the coordinate plane using a test point.
If the sign-flip rule still feels uncertain, redo Example 2 and verify your answer by substitution. To master linear inequalities with a teacher, explore Bhanzu's algebra tutor, algebra classes, or math tutoring. Want a live Bhanzu trainer to walk through inequalities with your child? Book a free demo class.
Read More
Linear Equations: Solving for a single value before moving to inequalities
Absolute Value: The step before absolute-value inequalities
Increasing and Decreasing Intervals: reading solution ranges off a graph
Was this article helpful?
Your feedback helps us write better content