How to Solve Inequalities: Rules, the Flip, and Worked Examples

The seesaw settled it: $4>2$ times $-1$ gives $-4<-2$. Every linear case runs on the four moves below plus that one guard — where how to solve inequalities departs from equations.

1. Simplify each side. Distribute, combine like terms, clear fractions.
2. Add or subtract the same number on both sides — the direction never changes.
3. Multiply or divide by a positive number — the sign keeps its direction.
4. Multiply or divide by a negative number, and the sign flips — $<$ becomes $>$, $\le$ becomes $\ge$.

How to solve inequalities, in one slogan: run the equation moves, flip once at step 4 — the answer is a stretch of numbers, $x>4$ meaning everything past 4.

Multiply $4>2$ by $-1$: the left becomes $-4$, the right $-2$ — smaller now. Without the flip we would claim $-4>-2$, clearly false — the first lesson in how to solve inequalities: the flip keeps truth true.

Smith and Gale (1904) prove the rule in two lines: "Thus $3<5$, but $-3>-5$. Therefore changing signs throughout an inequality reverses the inequality sign."

Positives flip nothing: $4>2$ times 3 is $12>6$. So how to solve inequalities comes down to one sentence: solve as you would an equation; reverse the sign exactly when a negative multiplication or division touches both sides. Say that sentence, and how to solve inequalities has no secrets left.

Adding or subtracting the same number on both sides never disturbs order: from $-4<2$, subtract 3 and $-7<-1$ holds. Stated generally: if $a<b$, then $a-c<b-c$ and $a+c<b+c$ — the first moves of how to solve inequalities feel like equation work.

One-move case: solve $x+5>9$ — subtract 5, and $x>4$: how to solve inequalities has produced its answer; anything past 4 works, 4 fails.

How to solve inequalities starts with knowing when nothing special happens — addition and subtraction never flip anything.

Fractions add one preliminary move: clear the denominators first — multiply every term by the LCD, positive, so no flip. Clear $\frac{1}{3}a-\frac{1}{8}a>\frac{5}{24}a+\frac{3}{4}$ with 24 and the fractions vanish before any sign decision.

Worked small case — solve $\frac{x}{2}+1>4$ and watch how to solve inequalities shrug fractions off: subtract 1, $\frac{x}{2}>3$; multiply by 2, $x>6$.

Fractions, in short, keep the standard order: LCD first, flip check last.

When the moves end, the answer is a stretch of the number line: $x>4$ an open dot at 4, arrow right; $x\le 1$ a closed dot, arrow left. Interval notation compresses it into $(4,\infty)$ and $(-\infty,1]$ — parenthesis excludes, bracket includes. How to solve inequalities reads its answers three ways, and the next table lines them up.

Division of labor: this page owns how to solve inequalities — moves and flip; drawings belong to the graph-reading page.

Checking belongs here: test one number inside the range, one outside — if both pass, how to solve inequalities has done its job here.

How to solve inequalities here means linear, one variable; neighbors live on their own pages.

• Graphing: solving and graphing inequalities, how to graph an inequality, and how to graph solutions to inequalities on a number line — the graphing pages.
• Absolute value: how to solve absolute value inequalities, how to solve absolute value, and how to do absolute value — its own page.
• Compound sentences: what does the word disjunction mean? An or-compound of two intervals — the compound page.
• Tools: a query like inequality cal wants the inequality calculator.
• Definition: the inequality mathematical definition, or definition of inequality in mathematics, is one line — two expressions compared by $<$, $>$, $\le$, $\ge$: the line how to solve inequalities protects on every page.

Linear and bar-free? This page is all you need.

One solution set, three costumes: the table below is where how to solve inequalities lands in notation — it always ends in one of these rows.

Pick the costume the question asks for; how to solve inequalities dresses its answers three ways, and how to solve inequalities always ends in one of these rows.

Problem. Solve $9y<54$ — a first look at how to solve inequalities — then test the edge.

Step 1. Divide both sides by 9 — positive, so how to solve inequalities keeps the sign where it started: $y<6$.

Step 2. Test the edge: $y=5$: $45<54$, true; $y=7$: $63<54$, false.

Note. The one-step case $7y<42$ shows the same shape: no flip, no fractions; even at its simplest, how to solve inequalities deserves its edge test.

Problem. Solve $7-2x>19$.

Step 1. Subtract 7: $-2x>12$.

Step 2. Divide by $-2$ — a negative number — so the guard fires and how to solve inequalities flips the sign: $x<-6$.

Check. $x=-7$: $7-2(-7)=21$ and $21>19$ holds; $x=-5$: $7+10=17$ and $17>19$ fails — checking is the part of how to solve inequalities that catches slips.

Note. The one-step case $-2x>4$ flips the same way, to $x<-2$ — see $4>2$. The flip lands mid-problem, the spot where how to solve inequalities usually goes wrong.

Problem. Solve $\frac{1}{2}x-\frac{1}{5}x>\frac{1}{2}$ — and watch how to solve inequalities take fractions.

Step 1. Multiply every term by 10, the LCD: $5x-2x>5$.

Step 2. Combine: $3x>5$.

Step 3. Divide by 3, positive: $x>\frac{5}{3}$ — no flip, exactly as how to solve inequalities treats positives.

Check. $x=2$: $1-\frac{2}{5}=\frac{3}{5}$, and $\frac{3}{5}>\frac{1}{2}$, true. $x=1$: $\frac{3}{10}>\frac{1}{2}$, false.

Note. Harder fraction cases clear the same way, with a larger LCD. The order here is exactly how to solve inequalities: clear, combine, decide the sign.

1. Skipping the flip. Divide $-2x>12$ by $-2$ and keep $>$: you get $x>-6$, full of failures. Flip once: $x<-6$. The flip is the whole gap between how to solve inequalities and how to solve equations.

2. Flipping after addition or subtraction. Subtracting 5 from $x+5>9$ changes no direction; only multiplication and division can flip — in how to solve inequalities, the flip comes last, on negatives only.

3. Flipping for positive multipliers. $3x>12$ divided by 3 stays $x>4$ — save the guard for negatives, or how to solve inequalities flips answers for nothing.

4. Treating the endpoint as included. $x>4$ excludes 4 — test it: $4>4$ is false. Open dot, parenthesis; brackets reserved for $\le$ and $\ge$ — half of how to solve inequalities comes down to honest boundaries.

5. Handing back one number. Solving produces a set: $y<6$ admits 5, 0, $-3$ and more — a lone "$y=5$" misses what how to solve inequalities means.

6. Trusting without a test. One inside and one outside value catch every sign slip — untested answers leave how to solve inequalities half-finished.