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Expression of IdeasTransitions

Jazz pianist Mary Lou Williams composed hundreds of works. ______ she wrote arrangements for bands led by Duke Ellington and Benny Goodman.

AHowever,Your pick

BIn addition,Correct

Why: the second sentence adds another accomplishment. Nothing is being contrasted.

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MathProblem-Solving and Data AnalysisHard

A store raises the price of a jacket by 20%. Later, it lowers the new price by 20%. The final price is what percent of the original price?

A80%

B96%Correct

C~~100%~~The trap

D104%

Check with $100: $100then$120then$96

Turn each change into a multiplier.
Up 20% means × 1.20. Down 20% means × 0.80.
Apply them in order.
1.20 × 0.80 = 0.96
Read the answer.
The final price is 96% of the original. That’s B.
Why not 100%?
The 20% cut is taken from the higher price, so it removes more than the 20% increase added.

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The Daily, plus 36 practice questions every day: 12 easy, 12 medium and 12 hard, each with a full explanation. Full-length Test A, Challenge Test 1 and all 1,023 vocabulary words are free too. No card needed.

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98 questions across four modules, in the digital SAT’s order: Reading & Writing Module 1 and Module 2 (27 questions each), a 10-minute break, then Math Module 1 and Module 2 (22 questions each). On official timing that’s 2 hr 14 min of testing. You get estimated section scores and a total at the end.

Yes. In the standard full-length tests, Module 2 of each section is the harder version if you get 60% or more of Module 1 right, and the easier version if you don’t. Challenge tests use the hardest material throughout.

Official timing, extended 1.5× time, or untimed, on every test, including the free ones. Extended time allows 48 minutes for each Reading & Writing module and about 52 for each Math module. Untimed removes the clock so you can focus on accuracy.

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Equivalent expressions · Lesson 4

Rational expressions

A rational expression is a fraction with polynomials on the top and bottom. You handle it the way you handle number fractions.

Simplify by factoring the top and bottom completely, then canceling the factors they share: $\frac{x^2 - 25}{x^2 + 2x - 15} = \frac{(x - 5)(x + 5)}{(x - 3)(x + 5)} = \frac{x - 5}{x - 3}$ . You may cancel only factors, whole pieces that multiply. You can't cancel the $x^2$ in $\frac{x^2 - 25}{x^2 + 2x - 15}$ , because $x^2$ is a term being added.

Canceling can hide a restriction. The original fraction is undefined at $x = 3$ and $x = -5$ , but $\frac{x - 5}{x - 3}$ is defined at $x = -5$ . The two are equal wherever the original is defined. SAT questions usually state a condition such as *for $x > 3$ * so this doesn't matter; if one asks you to solve an equation, a value that makes an original denominator 0 can't be a solution.

Add or subtract over a common denominator. Factor the denominators, build the least common denominator, and rewrite each fraction with it. When you subtract, put the whole second numerator in parentheses: $\frac{3}{x} - \frac{x - 1}{x + 2} = \frac{3(x + 2) - x(x - 1)}{x(x + 2)}$ .

Rewrite as a whole part plus a remainder by writing the numerator in terms of the denominator: $\frac{2x + 7}{x + 3} = \frac{2(x + 3) + 1}{x + 3} = 2 + \frac{1}{x + 3}$ .

Worked example Easy

Which expression is equivalent to $\frac{x^2 - 9}{x^2 + 5x + 6}$ for $x > 0$ ?

$\frac{x - 3}{x + 2}$ Answer
$\frac{-9}{5x + 6}$
$\frac{x + 3}{x + 2}$
$-\frac{3}{2}$

How to solve it

Factor the top as a difference of squares: $(x - 3)(x + 3)$ .
Factor the bottom: $(x + 2)(x + 3)$ , since $2 \cdot 3 = 6$ and $2 + 3 = 5$ .
Cancel the shared factor $x + 3$ : $\frac{x - 3}{x + 2}$ .

Why each choice is right or wrong

A. Correct. $\frac{(x - 3)(x + 3)}{(x + 2)(x + 3)} = \frac{x - 3}{x + 2}$ after canceling the shared factor $x + 3$ .
B. Incorrect. This cancels the $x^2$ terms, but $x^2$ is a term being added, not a factor. Check at $x = 1$ : the original is $\frac{-8}{12} = -\frac{2}{3}$ , while this choice is $-\frac{9}{11}$ .
C. Incorrect. This factors $x^2 - 9$ as $(x + 3)(x + 3)$ and then cancels one $x + 3$ . But $(x + 3)^2 = x^2 + 6x + 9$ .
D. Incorrect. This cancels the $x$ in $\frac{x - 3}{x + 2}$ to get $\frac{-3}{2}$ . That $x$ is a term, not a factor of the top and bottom, so it can't be canceled.

Worked example Medium

Which expression is equivalent to $\frac{5}{x - 2} - \frac{x + 8}{x^2 - 4}$ for $x > 2$ ?

$\frac{4x + 2}{x^2 - 4}$ Answer
$\frac{4x + 18}{x^2 - 4}$
$\frac{-x - 3}{x^2 - 4}$
$\frac{4x - 6}{x^2 - 4}$

How to solve it

Factor the second denominator: $x^2 - 4 = (x - 2)(x + 2)$ . That is the common denominator.
Rewrite the first fraction: $\frac{5(x + 2)}{(x - 2)(x + 2)}$ .
Subtract the whole second numerator: $\frac{5(x + 2) - (x + 8)}{x^2 - 4} = \frac{5x + 10 - x - 8}{x^2 - 4}$ .
Combine: $\frac{4x + 2}{x^2 - 4}$ .

Why each choice is right or wrong

A. Correct. Over the common denominator $x^2 - 4$ , the top is $5(x + 2) - (x + 8) = 5x + 10 - x - 8 = 4x + 2$ .
B. Incorrect. This subtracts only the $x$ from the second numerator: $5x + 10 - x + 8 = 4x + 18$ . The minus sign applies to all of $x + 8$ .
C. Incorrect. This changes the first denominator to $x^2 - 4$ without multiplying the 5 by $x + 2$ , so the top becomes $5 - x - 8 = -x - 3$ . Whatever multiplies the bottom must multiply the top.
D. Incorrect. This writes $5(x + 2)$ as $5x + 2$ , so the top becomes $5x + 2 - x - 8 = 4x - 6$ . The 5 multiplies the 2 too.

Worked example Hard

Which expression is equivalent to $\frac{6x + 11}{2x + 3}$ for $x > 0$ ?

$3 + \frac{2}{2x + 3}$ Answer
$3 + \frac{11}{2x + 3}$
$3 + \frac{11}{3}$
$3 - \frac{2}{2x + 3}$

How to solve it

Write the numerator in terms of the denominator: $3(2x + 3) = 6x + 9$ , so $6x + 11 = 3(2x + 3) + 2$ .
Split the fraction: $\frac{3(2x + 3)}{2x + 3} + \frac{2}{2x + 3} = 3 + \frac{2}{2x + 3}$ .
Check at $x = 1$ : $\frac{17}{5} = 3.4$ and $3 + \frac{2}{5} = 3.4$ .

Why each choice is right or wrong

A. Correct. $6x + 11 = 3(2x + 3) + 2$ , so the fraction is $3 + \frac{2}{2x + 3}$ . At $x = 1$ both forms equal 3.4.
B. Incorrect. This divides $6x$ by $2x$ to get 3 and keeps all of 11 as the remainder. But $3(2x + 3)$ uses up 9 of the 11, so only 2 is left. At $x = 1$ this choice is $3 + \frac{11}{5} = 5.2$ , not 3.4.
C. Incorrect. This divides each term on top by a term on the bottom, $\frac{6x}{2x}$ and $\frac{11}{3}$ . A fraction can't be split across the terms of its denominator.
D. Incorrect. This subtracts the wrong way, $9 - 11 = -2$ . The remainder is $11 - 9 = 2$ . At $x = 1$ this choice is $3 - \frac{2}{5} = 2.6$ , not 3.4.

Trap Canceling terms instead of factors

In $\frac{x + 6}{x + 2}$ nothing cancels: the $x$ 's are terms of sums. Cancel only after factoring, and only whole factors. A quick test catches a bad cancel: substitute a number such as $x = 1$ into the original and into your answer. If the values differ, your answer is wrong.

Trap Subtracting only the first term

When you subtract a fraction, its whole numerator is subtracted. Write it in parentheses, such as $-(x + 8)$ , before you distribute. The choice with $+8$ instead of $-8$ is the trap.

Desmos When Desmos is faster

Graph the original as $y =$ and each choice on its own line. Type fractions with the slash key. The equivalent choice lies on top of the original curve wherever the original is defined.

If the question says *for $x > 2$ *, compare the graphs only to the right of $x = 2$ . A simplified form can be defined at a value where the original isn't, and that single point doesn't make it wrong.

You can also compare numbers: type the original with a value such as 3 in place of $x$ , then each choice the same way. A choice with a different value is wrong. One matching value doesn't prove equivalence, so if two choices match, try a second value.

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Each test mirrors the digital SAT: 98 questions across four modules, 2 hr 14 min on official timing.

A little harder than the real SAT. These tests can run slightly tougher than test day, which makes them good practice: if you can handle these, the real one should feel easier. No question appears in more than one test, Challenge Tests included.

Challenge Tests Hardest of the hardest

Same 98-question format, but every question is drawn from the toughest SAT material: Challenging-tier math throughout and the hardest reading.

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Study the classic SAT vocabulary list. Browse, flip flashcards, or quiz yourself.

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Level up your SAT skills.

The real test, rehearsed.

One run a day. One skill at a time.

Play the Daily.

Miss one, learn why.

Drill that skill.

Beat your best.

Built around the real SAT blueprint.

Free daily practice

Real SAT timing

Adaptive full-length tests

Vocabulary

Your progress

Games

Every answer, explained.

Turn each change into a multiplier.

Apply them in order.

Read the answer.

Why not 100%?

What test day looks like.

Reading & Writing · Module 1

Reading & Writing · Module 2

Break

Math · Module 1

Math · Module 2

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Is Peakscor affiliated with the College Board?

Your next point starts today.

Rational expressions

Worked example Easy

How to solve it

Why each choice is right or wrong

Worked example Medium

How to solve it

Why each choice is right or wrong

Worked example Hard

How to solve it

Why each choice is right or wrong

Trap Canceling terms instead of factors

Trap Subtracting only the first term

Desmos When Desmos is faster

Your progress

Practice

Full-length tests

Challenge Tests Hardest of the hardest

Vocab

Free every day. Pro when you're serious.

FREE

PRO

PEAK