Digital SAT · Math · Unit 1 · Algebra

Lesson 1.1 · Expressions

On the test: about 2–4 questions a test, almost all Easy or Medium — free marks if you have the reflexes

Big idea. Two expressions are EQUIVALENT when they give the same number for every value of the letters. That one sentence is three tools: multiply it out (algebra), put in an easy number (the fast lane), or graph both and see ONE curve (Desmos).

🎛 Move it · one curve or two?

The target is from Official Practice Test 9 (Module 1, Q5). Pick each option and drag $x$ . The right option draws the SAME curve as the target; a wrong one peels away somewhere — and the place it peels away is a number that kills it.

Target: $9 x^{2} + 5 x$

drag x: 1

target 14A 14

✓ One curve — the graphs lie on top of each other everywhere. Equivalent.

🧑‍🏫 Simplify it like a tutor wouldstep 1 of 5

1
📋 The question · SAT · June 2024 · Math 800 Book Q30
Which expression is equivalent to $(x^{3} + 6 x^{2} - 8 x) + 8 (x^{2} + 4)$ ?
1. A) $x^{3} + 4 x^{2} - 8 x + 32$
2. B) $x^{3} + 14 x^{2} - 8 x + 32$
3. C) $x^{3} + 8 x^{2} - 8 x + 32$
4. D) $x^{3} + 6 x^{2} - 8 x + 32$
the question
$(x^{3} + 6 x^{2} - 8 x) + 8 (x^{2} + 4)$
A real June 2024 question. Two jobs hide in it: a bracket with a number in front, and terms that can be merged.

Why it works · why “like terms” only

6 x^{2} + 8 x^{2}

is 6 squares plus 8 squares = 14 squares. But

6 x^{2} + 8 x

is squares plus lines — different things, like 6 apples plus 8 pears. They stay apart.

Trap · the square of a sum

(a + b)^{2}

is NOT

a^{2} + b^{2}

. It is

a^{2} + 2 ab + b^{2}

— the middle term

2 ab

is the one the wrong option forgets (March US 2023:

(\frac{a}{4} + \frac{b}{3})^{2}

Fast lane · check only the ends

The options usually differ in one place. Work out just the highest-power term and the constant (e.g.

3 x^{3} \cdot 5 x^{2} = 15 x^{5}

and

4 \cdot 8 x = 32 x

) — often that alone leaves one option standing.

Desmos move · graph both

Type the original as y = … and an option as another y = …. One curve = equivalent. A second letter (like

y

) becomes a slider in Desmos — or just set it to 2.

Real question 1SAT · March US 2023 · Math 800 Book Q2

Which of the following expressions is equivalent to $(\frac{a}{4} + \frac{b}{3})^{2}$ ?

Real question 2SAT · March 2024 · Math 800 Book Q22

Which expression is equivalent to $\frac{y + 2}{x - 9} + \frac{y ( x - 9 )}{x ^{2} y - 9 x y}$ ?

🎛 Move it · peel the letter

From the March 2024 test. An equation is a balance — do the same to both sides. Pick the move that takes the next layer OFF $w$ .

Get w alone on one sidemove 1 of 2

v = - \frac{w}{161 x}

🧑‍🏫 Layers come off in reverse orderstep 1 of 5

1
the question
$ak = \frac{b}{12} (7 + m)$
June 2024: express $m$ in terms of $a$ , $b$ , $k$ . First ask: what was done TO $m$ , in what order?

Why it works · why “both sides”

An equation says two things are equal. Do the same thing to both and they stay equal — that is the only move algebra has.

Fast lane · divide by the reciprocal form

\frac{20}{p} = \frac{20}{q} - \frac{20}{r} - \frac{20}{s}

: divide everything by 20 FIRST. Then

\frac{1}{q} = \frac{1}{p} + \frac{1}{r} + \frac{1}{s}

and

q

is the flip. (Aug 2024 and Oct 2024 both asked this.)

Trap · the letter on both sides

P C = N (44 - C)

C

is on both sides. Collect every

C

on one side and factor it out — an answer "for

C

" can never contain

C

, which kills two options at once.

Fast lane · pick numbers

Choose easy values for the other letters, work out the wanted letter from the GIVEN equation, then test the options. In

v = - \frac{w}{161 x}

w = 161

x = 1

gives

v = - 1

; only

- 161 v x = 161

✓.

Real question 3SAT · August 2024 · Math 800 Book Q42

\frac{20}{p} = \frac{20}{q} - \frac{20}{r} - \frac{20}{s}

The given equation relates the positive variables $p$ , $q$ , $r$ , and $s$ . Which of the following is equivalent to $q$ ?

Real question 4SAT · December 2024 · Math 800 Book Q64

P C = N (44 - C)

The given equation relates the variables $P$ , $N$ , and $C$ . Which expression represents the value of $C$ for distinct positive values of $P$ and $N$ ?

🎛 Move it · spot the chunk

December 2023: $9 (x + 3) = 81$ , what is $x + 3$ ? Tap the chunk the question wants.

The question asks for $x + 3$ . Tap where that chunk already sits in the equation.

🧑‍🏫 Name the chunk, then it is one letterstep 1 of 5

1
📋 The question · SAT · March 2024 · Math 800 Book Q25
$8 + \frac{9 ( 6 - x )}{4} = \frac{7 ( 6 - x )}{3}$
What is the value of $6 - x$ ?
(type the answer)
the question
$8 + \frac{9 ( 6 - x )}{4} = \frac{7 ( 6 - x )}{3}$
March 2024: what is the value of $6 - x$ ? The chunk $6 - x$ appears twice. Never open those brackets.

Fast lane · read the LAST line first

The question tells you its target:

4 x - 4

7 p

x - 9

x^{6} - y^{6}

. Look for it — or a multiple of it — inside what you are given.

16 x + 8 = 4 (4 x + 2)

;

x^{6} - y^{6} = (x^{3} + y^{3}) (x^{3} - y^{3})

Trap · answering x

The wrong options almost always include

x

itself (6 in "

9 (x + 3) = 81

", 49 in "

\frac{6}{7} p + 12 = 54

, find

7 p

"). Solving for

x

is fine — stopping there is the trap.

Desmos move · graph both sides

Type the left side as y = … and the right side as y = …. Tap the crossing: its

x

is the solution. Then do the last step (×7, −4, …) on the calculator.

Why it works · why a chunk can be treated as a letter

x + 3

" is just a number we have not met yet. Giving it a name (

u

) does not change it — it only stops you from taking it apart and rebuilding it.

Real question 5SAT · June 2023 · Math 800 Book Q5type the answer

If $\frac{6}{7} p + 12 = 54$ , what is the value of $7 p$ ?

Real question 6SAT · October 2024 · Math 800 Book Q57

If $\frac{42}{x} = 7 x$ , what is the value of $7 x^{2}$ ?

🎛 Move it · the pick-numbers machine

August US 2024: a tree grows 51 cm every $m$ months — how much in $k$ years? Choose numbers that make the arithmetic easy (try $m = 12$ , $k = 1$ : then the answer is plainly 51). Watch which options survive.

m =12k =1

True value with these numbers: 5151 cm every 12 months is $\frac{51}{12}$ cm a month; 1 year is 12 months.

\frac{17 m}{4 k}

\frac{17 k}{4 m}

0.3542

\frac{612 m}{k}

7344

\frac{612 k}{m}

2 options survive — change a number and test again.

🧑‍🏫 Words to maths, one phrase at a timestep 1 of 5

1
📋 The question · SAT · March US 2026 (Form A) · Module 1 · Q11
The number $b$ is 15 more than half of the number $a$ . Which equation represents the relationship between $a$ and $b$ ?
1. A) $b = 30 a$
2. B) $b = 15 a + 2$
3. C) $b = \frac{a}{15} + 2$
4. D) $b = \frac{a}{2} + 15$
phrase
$"half of a " \to \frac{a}{2}$
March US 2026: " $b$ is 15 more than half of $a$ ." Translate one phrase at a time.

Fast lane · pick numbers

When every option is letters, pick numbers that make the TRUE answer obvious (

m = 12

months = 1 year). Then test the options. If two survive, change one number.

Trap · the swapped option

Rate questions always offer the right numbers with the letters upside down (

\frac{612 m}{k}

). Sanity check: more months per 51 cm must mean LESS growth, so

m

belongs on the bottom.

Why it works · why units work

A rate is a fraction of units. Multiplying by a time cancels the time unit exactly like cancelling a number — so the units tell you whether to multiply or divide.

Desmos move · inequalities from words

Turn "at most" into

\leq

, graph both sides, and read off where one curve is below the other.

Real question 7SAT · November 2024 · Math 800 Book Q62

One gallon of sealant costs $23 and will cover 250 square feet of a surface. A deck has a total surface area of $d$ square feet. Which of the following represents the cost $c$ , in dollars, of the sealant needed to cover the deck twice?

Real question 8SAT · October US 2024 · Math 800 Book Q52type the answer

In a set of four consecutive odd integers, where the integers are ordered from least to greatest, the first integer is represented by $x$ . The product of 28 and the third odd integer in the set is at most the value of 50 less than the sum of the first and fourth odd integers in the set. What is the greatest possible value of $x$ ?

🧑‍🏫 Rule · Multiply two brackets: every term by every term (FOIL)step 1 of 4

1
📋 The question · Gen Z Study
Which expression is equivalent to $(2 x + 3) (x - 5)$ ?
1. A) $2 x^{2} - 7 x - 15$
2. B) $2 x^{2} - 15$
3. C) $2 x^{2} + 7 x - 15$
4. D) $3 x - 2$
the plan
$(2 x + 3) (x - 5)$
Each term of the first bracket multiplies each term of the second: 2 × 2 = 4 products.

🧑‍🏫 Rule · Three special products to know by heartstep 1 of 4

1
square of a sum
$(a + b)^{2} = a^{2} + 2 ab + b^{2}$
Never forget the middle $2 ab$ .

🧑‍🏫 Rule · Factor out the greatest common factor, then cancelstep 1 of 3

1
📋 The question · Gen Z Study
For $x \neq = 3$ , which expression is equivalent to $\frac{4 x ^{2} - 12 x}{2 x - 6}$ ?
1. A) $2 x$
2. B) $x - 3$
3. C) $2 x - 6$
4. D) $4 x$
factor the top
$4 x^{2} - 12 x = 4 x (x - 3)$
The biggest number and the lowest power of $x$ that divide every term.

Take with you

▸Equivalent = same value for EVERY input = the SAME curve on the graph.
▸Distribute to every term; merge only like terms (same letters, same powers); $(a + b)^{2}$ has a middle term $2 ab$ .
▸Rearranging: list what was done to the letter, then undo it in reverse order. An answer “for $C$ ” never contains $C$ .
▸Read the last line first — the question’s target ( $7 p$ , $x - 9$ , $4 x - 4$ ) is usually a chunk already sitting in the equation.
▸Words: translate phrase by phrase; write units on rates; “twice” doubles before you divide.
▸When the options are all letters: pick easy numbers and test. When there is an equation: graph both sides and tap the crossing.

Keep reading — this lesson marks itself done automatically.