Mistake Master
Four sign posts, two boundary reads
Sign errors are not random; they happen at four specific posts, and boundary errors at two specific reads. Minus-a-negative, the distributed minus, the squared negative, the flipped inequality; then the direction word and the endpoint. Name the posts and the vague instruction to "watch your signs" becomes six concrete checkpoints.
§1
What this topic is about
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Signs and boundaries fail at a short, predictable list of posts, and the SAT writes a distractor for each. This topic names the posts: subtracting a negative, distributing a minus, squaring a negative, dividing an inequality by a negative, and on the boundary side, at least/at most direction and inclusive-versus-exclusive endpoints.
§2
The four sign posts
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Almost every sign error on the test happens at one of four moments. Knowing the posts converts a vague "be careful with negatives" into four specific checkpoints.
- Minus a negative ADDS: $7 - (-5) = 12$.
- A distributed minus reaches EVERY term: $5 - 2(x - 3) = 11 - 2x$.
- A squared negative is positive: $(-4)^2 = 16$; only $-(4^2)$ is $-16$.
- Dividing an inequality by a negative FLIPS the symbol.
Worked example. What is $x^2 - 3x$ when $x = -2$?
$(-2)^2 = 4$ and $-3(-2) = +6$.
$4 + 6 = 10$. Keeping a minus on the second term gives $-2$; putting one on the square gives $-10$.
§3
Direction words
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"At least," "at most," "more than," and "no more than" each pin both a direction and an endpoint treatment. Translating them by feel is how the symbol ends up backward.
- At least $n$: $x \geq n$. At most $n$: $x \leq n$. Both INCLUDE the boundary.
- More than / fewer than are strict: the boundary itself fails.
- Translate the phrase before touching the algebra, not after.
Worked example. A rider must be at least $12$ years old. Which inequality?
At least means $12$ works and everything above does: $a \geq 12$.
A $12$-year-old may ride, so the symbol must include $12$; $a > 12$ wrongly turns them away.
§4
Counting between bounds
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Counting integers in an interval is a two-endpoint decision: each endpoint is admitted or excluded by ITS OWN symbol. Most wrong counts differ from the right one by exactly the mishandled endpoints.
- Check each endpoint against its own symbol, separately.
- Strict $<$ excludes; $\leq$ includes.
- "Strictly between" excludes both ends.
Worked example. How many integers satisfy $-3 < n \leq 2$?
The $<$ excludes $-3$; the $\leq$ admits $2$.
$-2, -1, 0, 1, 2$: five. Admitting $-3$ gives $6$; dropping $2$ gives $4$.
§5
Boundaries in word problems
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Budget and capacity problems end on a boundary decision: can the exact boundary value be taken? "At most" says yes; "fewer than" says no. The last step of the algebra is a words step.
Slow down at the four sign posts, translate direction words before the algebra, judge each endpoint by its own symbol, and let the problem's words decide whether the boundary itself is allowed.
§6
Two patterns that cost real points
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Two patterns cover this topic. They are the same ones the diagnostic routes on.
A sign falls at a predictable post.
Minus-a-negative collapses to subtraction, a distributed minus stops at the first term, a squared negative comes out negative, or a negative divisor leaves the inequality unflipped.
Fix. Treat the four posts as checkpoints: name the post you are at, apply its rule deliberately, and only then continue the arithmetic.
A boundary gets the wrong treatment.
"At least" points the symbol down, a strict endpoint sneaks into a count, or an included endpoint gets dropped at the finish.
Fix. Translate the phrase first and test the boundary value in words: does exactly $n$ satisfy the story? The answer decides the symbol and the count.
Ten quick checks across the patterns: all four sign posts, direction-word translation, endpoint-aware counting, absolute-value branches, and budget boundaries. Pick or type your answer, then check. Progress is saved.