Check what you know. Brush up on the big ideas.
Big ideas
Do this: Read one idea, then explain it in your own words.
To evaluate an expression, replace each variable with its complete value. Parentheses protect a negative input before applying powers or subtraction.
Calculate grouping and powers before multiplication or division, then addition or subtraction. Work left to right within the same priority.
The distributive property gives . A minus sign outside parentheses multiplies every inside term by −1.
Two expressions are equivalent when they agree at every input where both are defined. One matching example can support a check but cannot prove equivalence.
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The value is 15.
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The distribution and collection of like terms establish equivalence for all real x.
Do this: Explain why the tempting shortcut fails.
Wrong idea: . The factor multiplies every term inside: . Substituting exposes the error because the two proposed forms give 10 and 5.
Worked example
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Worked example
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Big ideas
Do this: Read one idea, then explain it in your own words.
For a positive whole-number exponent n, means n factors of a multiplied together. Parentheses show whether a sign or coefficient belongs to the base.
A product of sums needs every term from one factor multiplied by every term from the other. Collect like terms only after distributing.
The identity comes from multiplying two complete binomials. A square does not distribute over addition.
Substitution at a convenient input can catch a missed term. A justified expansion uses distribution, not just one numerical check.
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The middle term comes from two cross-products.
Do this: Explain why the tempting shortcut fails.
Wrong idea: . Squaring means multiplying (x+5)(x+5), so the two cross-products add . The full expansion is .
Worked example
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Worked example
Do this: Predict the next step on paper, then compare the worked reasoning.
Big ideas
Do this: Read one idea, then explain it in your own words.
An equation keeps the same solutions when you add or subtract the same quantity on both sides, or multiply or divide both sides by the same nonzero constant.
Distribute carefully and combine like terms. Moving a term is shorthand for a balancing operation, not a rule that changes signs without reason.
Multiplying or dividing an inequality by a negative number reverses its direction. Adding the same number to both sides does not reverse it.
A bracket includes a finite endpoint; a parenthesis excludes it. Use parentheses with infinity. An equation can have one solution, no solution or every real input as a solution.
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Both sides agree.
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Input −3 makes equality; input −4 gives .
Do this: Explain why the tempting shortcut fails.
Wrong idea: dividing by −3 leaves . A negative divisor reverses the order, giving . A test input such as confirms the corrected inequality.
Worked example
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Worked example
Do this: Predict the next step on paper, then compare the worked reasoning.
Then , so . The interval is .
