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98 questions across four modules, in the digital SAT’s order and on its timing, with a highlighter, a question map and a timer you can see.

A Peakscor full-length test in progress: Reading and Writing, Test A, Module 1 of 4, question 7 of 27, with the passage, four answer choices, a highlighter, a 24:18 timer and Flag, Back and Next buttons.

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Every question is tagged by section, domain, skill and difficulty, so you can practice exactly what you need. Full tests run like test day: two Reading & Writing modules, a break, then two Math modules.

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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
C100%The trap
D104%

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

  1. Turn each change into a multiplier.

    Up 20% means × 1.20. Down 20% means × 0.80.

  2. Apply them in order.

    1.20 × 0.80 = 0.96

  3. Read the answer.

    The final price is 96% of the original. That’s B.

  4. 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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Every Peakscor full test follows this order. In the standard tests, the second module of each section adapts to how you did on the first.

  1. Reading & Writing · Module 1

    27 questions32 min

  2. Reading & Writing · Module 2

    27 questions32 min

    Adapts to how you did on Module 1.
  3. Break

    10 min

  4. Math · Module 1

    22 questions35 min

  5. Math · Module 2

    22 questions35 min

    Adapts to how you did on Module 1.

98 questions · 2 hr 14 min of testing, plus the break

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The Daily, plus 36 practice questions every day: 9 easy, 9 medium, 9 hard and 9 challenging, each with a full explanation. Full-length Test A, Challenge Test 1 and all 1,023 vocabulary words are free too, as are the 20-question diagnostic, each Learn module’s overview and first lesson, and PEAK Survivor’s first map. No card needed.

Pro is $20 a month or $200 a year: unlimited practice with no daily limit, plus every full-length test and Challenge test, and every game map and level. Peak is $29 a month or $280 a year: everything in Pro, plus the full Learn course and Peak Plan. Cancel anytime from your account.

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.

A quick run with three hearts. Each right answer builds your streak: the questions get harder as it grows, and your points multiply (×2 from three in a row, ×3 from six). A miss costs a heart and drops you back to easy. Best is your longest streak.

Every day at midnight Central Time. You get a fresh 9 easy, 9 medium, 9 hard and 9 challenging.

No. Peakscor is an independent SAT practice tool, not affiliated with the College Board. Scores shown are estimates.

Your next point starts today.

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Nonlinear equations and systems · Lesson 4

Absolute value equations

The absolute value ∣A∣|A| is the distance of AA from 0, so it is never negative. ∣A∣=4|A| = 4 means AA is 4 units from 0: A=4A = 4 or A=−4A = -4.

Isolate the absolute value first. In 2∣x−3∣+1=92|x - 3| + 1 = 9, subtract 1 and divide by 2 before touching the bars: ∣x−3∣=4|x - 3| = 4.

Then split into two cases. x−3=4x - 3 = 4 gives x=7x = 7, and x−3=−4x - 3 = -4 gives x=−1x = -1.

Count before you solve. Once ∣A∣=c|A| = c is isolated, with cc a number: if c>0c > 0, there are two cases (and, when AA is linear in xx, two solutions). If c=0c = 0, there is one case, A=0A = 0 (one solution when AA is linear). If c<0c < 0, there are none, since a distance can't be negative.

When xx also appears outside the bars, as in ∣x−4∣=2x+1|x - 4| = 2x + 1, solve both cases and check each answer in the original. A case can produce a value that fails. Here x−4=2x+1x - 4 = 2x + 1 gives x=−5x = -5, but then the right side is −9-9, and an absolute value can't equal a negative, so x=−5x = -5 fails. The other case, x−4=−(2x+1)x - 4 = -(2x + 1), gives 3x=33x = 3 and x=1x = 1, which checks: ∣−3∣=3=2(1)+1|-3| = 3 = 2(1) + 1. The only solution is x=1x = 1.

On a graph, y=∣x−3∣y = |x - 3| is a V with its corner at (3,0)(3, 0). The solutions of ∣x−3∣=4|x - 3| = 4 are where the V meets the horizontal line y=4y = 4.

The graphs of y = |x - 3| and y = 4−22468−2−112345678xy(-1, 4)(7, 4)(3, 0)
The graphs of y=∣x−3∣y = |x - 3| and y=4y = 4 A V-shaped graph with its corner at (3, 0), opening upward, and a horizontal line at y = 4. They meet at (-1, 4) and (7, 4), so |x - 3| = 4 has the solutions -1 and 7.

Worked example Easy

∣x+2∣=6|x + 2| = 6

What are all the solutions to the given equation?

  1. x=4x = 4 only
  2. x=−4x = -4 and x=4x = 4
  3. x=−8x = -8 and x=4x = 4Answer
  4. x=−4x = -4 and x=8x = 8

How to solve it

  1. The absolute value is already alone. Split into two cases.
  2. x+2=6x + 2 = 6 gives x=4x = 4.
  3. x+2=−6x + 2 = -6 gives x=−8x = -8.
  4. Check: ∣6∣=6|6| = 6 and ∣−6∣=6|-6| = 6.

Why each choice is right or wrong

  • A. Incorrect. This solves only the positive case, x+2=6x + 2 = 6, which gives x=4x = 4 only. x+2x + 2 can also be −6-6, which gives x=−8x = -8.
  • B. Incorrect. This solves x+2=6x + 2 = 6 and then puts ±\pm on the answer. The ±\pm belongs on the 6: x+2=±6x + 2 = \pm 6. At x=−4x = -4, ∣−2∣=2|-2| = 2, not 6.
  • C. Correct. x+2=6x + 2 = 6 or x+2=−6x + 2 = -6, so x=4x = 4 or x=−8x = -8. Both give ∣x+2∣=6|x + 2| = 6.
  • D. Incorrect. This treats x+2x + 2 as x−2x - 2: x−2=±6x - 2 = \pm 6 gives 8 and −4-4. At x=8x = 8, ∣10∣=10|10| = 10, not 6.

Worked example Medium

3∣2x−1∣+4=193|2x - 1| + 4 = 19

What is the sum of the solutions to the given equation?

  1. −13-\frac{1}{3}
  2. 00
  3. 11Answer
  4. 33

How to solve it

  1. Isolate the absolute value. Subtract 4: 3∣2x−1∣=153|2x - 1| = 15. Divide by 3: ∣2x−1∣=5|2x - 1| = 5.
  2. Case 1: 2x−1=52x - 1 = 5, so 2x=62x = 6 and x=3x = 3.
  3. Case 2: 2x−1=−52x - 1 = -5, so 2x=−42x = -4 and x=−2x = -2.
  4. Sum: 3+(−2)=13 + (-2) = 1.

Why each choice is right or wrong

  • A. Incorrect. This splits into cases before isolating the bars: 3(2x−1)+4=±193(2x - 1) + 4 = \pm 19, which gives x=3x = 3 and x=−103x = -\frac{10}{3}, whose sum is −13-\frac{1}{3}. Isolate the absolute value first, then split.
  • B. Incorrect. This subtracts 1 instead of adding it in the second case: 2x=−5−12x = -5 - 1, so x=−3x = -3, and 3+(−3)=03 + (-3) = 0. Adding 1 gives 2x=−42x = -4 and x=−2x = -2.
  • C. Correct. ∣2x−1∣=5|2x - 1| = 5 gives x=3x = 3 or x=−2x = -2, and 3+(−2)=13 + (-2) = 1.
  • D. Incorrect. 3 is only the solution from the positive case. The negative case adds x=−2x = -2, and the sum is 1.

Trap Splitting before isolating

The two cases apply to the absolute value alone. In 3∣2x−1∣+4=193|2x - 1| + 4 = 19, writing 3(2x−1)+4=−193(2x - 1) + 4 = -19 is wrong. Clear everything outside the bars first, then split.

Trap An absolute value equal to a negative

∣x+1∣=−4|x + 1| = -4 has no solution, even though x+1=4x + 1 = 4 and x+1=−4x + 1 = -4 both give numbers. An absolute value is never negative. And ∣x+1∣=0|x + 1| = 0 has exactly one solution, x=−1x = -1, not two.

Desmos When Desmos is faster

Graph each side as its own y=y =. Type abs( or the | key for the bars: y=3∣2x−1∣+4y = 3|2x - 1| + 4 and y=19y = 19. Click the points where the graphs cross; their xx-coordinates are the solutions, here −2-2 and 3.

This is especially useful when xx is also outside the bars, as in ∣x−4∣=2x+1|x - 4| = 2x + 1: graph y=∣x−4∣y = |x - 4| and y=2x+1y = 2x + 1, and they cross only at (1,3)(1, 3). Every crossing on the graph is a real solution, so there is nothing to check afterward.

For a simple equation such as ∣x+2∣=6|x + 2| = 6, the two cases take seconds by hand.

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Full-length tests

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.

Vocab

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

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