Mistake Master
Solving inequalities and reading boundaries
A linear inequality describes a range of values, not a single point. You solve it almost exactly like an equation, with one extra rule: multiplying or dividing both sides by a negative flips the symbol. The points leak in three places, a symbol that should have flipped and didn't, a strict bound read as inclusive or the reverse, and the boundary handed in when the question asked for a value inside the range. Solve for the range, flip when you scale by a negative, then read off the exact thing the item names.
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What this topic is about
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A linear inequality is a linear equation with the equals sign swapped for $\lt$, $\gt$, $\le$, or $\ge$. Its solution is not one number but every number that makes the statement true, a range you can draw as a ray or a segment on the number line. The work splits in two: solve for the range, then report the exact quantity the question asks for. The arithmetic is short. The points leak when a symbol that should flip stays put, when a filled or open boundary is read the wrong way, and when the boundary itself is reported instead of a value the question actually wanted.
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Solving: isolate the variable, flip on a negative
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Solve an inequality the way you solve an equation: undo the additions and subtractions, then undo the multiplication or division. The one move with no equation counterpart is the flip. Whenever you multiply or divide both sides by a negative number, the symbol reverses direction. Forgetting that single step is the most common way points leak on this topic.
- Move constants first with addition and subtraction; those steps never flip the symbol.
- When you divide or multiply by a negative, reverse the symbol: $\lt$ becomes $\gt$, $\le$ becomes $\ge$.
- Keep the sign on the bound. Dividing by $-2$ makes the boundary negative, not positive.
Worked example. Solve $-2x + 4 \gt 10$.
Subtract $4$ from both sides: $-2x \gt 6$.
$$-2x \gt 6 \implies x \lt -3.$$ Dividing both sides by $-2$ reverses the symbol. Leaving it as $x \gt -3$ keeps the symbol pointing the wrong way, the single dropped step the whole wrong answer rides on.
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Compound inequalities: do every part, flip every part
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A compound inequality like $2 \le -x + 5 \lt 6$ pins a variable between two bounds. Whatever you do to the middle, do to all three parts at once. The flip rule still holds, and it hits both symbols together: multiplying through by a negative reverses the left symbol and the right symbol in the same move.
- Add or subtract across all three parts before you scale.
- When you multiply or divide every part by a negative, flip both symbols at once.
- Read the finished range left to right so the smaller bound sits on the left.
Worked example. Solve $2 \le -x + 5 \lt 6$.
$-3 \le -x \lt 1$.
$$3 \ge x \gt -1 \implies -1 \lt x \le 3.$$ Skipping the flip leaves $-3 \le x \lt 1$, a different range built on the missed step.
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Reading the boundary: strict or inclusive
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On the number line, the boundary point carries the information that tells strict from inclusive. A filled dot means the endpoint is part of the solution, an inclusive $\le$ or $\ge$. An open dot means the endpoint is left out, a strict $\lt$ or $\gt$. The direction of the shading tells you which way the inequality runs.
- Filled dot: the endpoint counts. The symbol is $\le$ or $\ge$.
- Open dot: the endpoint is excluded. The symbol is $\lt$ or $\gt$.
- Shading to the right means greater than the boundary; to the left means less than.
Worked example. What inequality does this number line show?
The dot on $2$ is filled, so $2$ is part of the solution.
The shading runs right, toward the larger numbers, so every value is at least $2$. That is $x \ge 2$. Reading the filled dot as open would give $x \gt 2$, dropping the endpoint the solid dot keeps in.
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Report the exact value the item asks for
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Solving the inequality is usually the short part. The trap is the last step. A question may ask for the greatest integer that works, the least value, or a specific quantity, and the boundary of the range is not always that answer. When the symbol is strict, the boundary itself is excluded, so the greatest integer sits one step inside the range.
Worked example. What is the greatest integer $x$ for which $2x + 1 \lt 9$?
Subtract $1$, then divide by $2$: $x \lt 4$.
$$x \lt 4 \implies \text{greatest integer } x = 3.$$ Because $\lt$ is strict, $4$ is not a solution, so the greatest integer that works is $3$. Handing in $4$ reports the boundary, a value the strict symbol leaves out.
Flip the symbol whenever you scale by a negative, match the dot to strict or inclusive, and report the exact value the question names, not the boundary when the symbol leaves it out.
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Three patterns that cost real points
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Three patterns recur on inequalities. They are the same ones the diagnostic routes on.
A symbol that should flip stays put.
Both sides get multiplied or divided by a negative, but the symbol keeps its original direction. $-2x \gt 6$ gets written as $x \gt -3$ instead of $x \lt -3$. Every step after that is clean, but the range now points the wrong way.
Fix. The moment you divide or multiply by a negative, reverse the symbol. A $\gt$ becomes a $\lt$.
A strict bound is read as inclusive, or the reverse.
An open dot is read as filled, or $\lt$ is treated as if it included the endpoint. The range is right, but one endpoint is wrongly kept in or left out, which changes which integers count.
Fix. Match the symbol to the dot. Filled with $\le$ or $\ge$ keeps the endpoint; open with $\lt$ or $\gt$ drops it.
The boundary is handed in instead of the value asked.
The range is solved correctly, then the boundary number is reported when the item asked for the greatest integer, a count, or a quantity inside the range. The number is real, it just answers a different question.
Fix. Underline what the item asks for before solving, then read that exact value off the finished range. A strict bound is not itself a solution.
Ten quick checks across the three patterns: flipping the symbol when you scale by a negative, telling a strict bound from an inclusive one, and reporting the exact value the item asks for. Pick or type your answer, then check. Progress is saved.