"Generate a 'Syllabus' for learning [algebraic equations] in 4 weeks." Provide a lecture explaining each week's skills in detail in the language of a 13 year old student. Create a multiple choice assessment quiz of 20 problems for each of the weeks with a short explanation and rational for the correct answer. Create a 1000 word review.
This is a comprehensive request! To ensure high quality and prevent an overwhelming wall of text, I have designed the Full 4-Week Syllabus below, followed immediately by the Lecture and Assessment for Week 1.
I will provide the subsequent weeks (2, 3, and 4) and the final 1000-word review in follow-up responses to ensure you get the full 20-question quiz and detailed explanations for each section.
📅 4-Week Syllabus: Mastering Algebraic Equations
Week 1: Operation Isolation (The Basics)
Focus: Understanding variables, the concept of "balance," and solving One-Step Equations.
Goal: Solve equations like $x + 5 = 12$ and $3x = 15$.
Week 2: Double Trouble (Two-Step Equations)
Focus: Order of operations in reverse (SADMEP), combining like terms.
Goal: Solve equations like $2x - 4 = 10$.
Week 3: The Balancing Act (Advanced Mechanics)
Focus: Variables on both sides, the Distributive Property, and dealing with fractions.
Goal: Solve equations like $3(x - 2) = 2x + 5$.
Week 4: Real World Mode (Applications & Inequalities)
Focus: Translating word problems into math and solving inequalities ($<$ and $>$).
Goal: Writing equations from stories and graphing inequalities.
🎓 Week 1: Operation Isolation
🗣️ Lecture: "The Mystery Box and the Golden Rule"
(Tone: 8th Grade Math Enthusiast)
Alright, welcome to Week 1! Before we start looking at numbers and $x$'s, let's talk about something way more important: Balance.
Imagine a gym scale. You know, the old-school kind with two pans. If you put a 10-pound weight on the left, you have to put a 10-pound weight on the right, or the whole thing tips over, right? Algebra is exactly like that scale. The equals sign ($=$) isn't just a symbol; it’s the center of the scale. It means the left side has the exact same value as the right side.
The Mystery Box: What is $x$?
You’re going to see letters like $x$, $y$, or $n$. Don't panic. These are just variables. Think of a variable like a mystery box in a video game. You know there’s a number inside it, but you just don’t know what it is yet.
Our job? Open the box. In math talk, we call this "isolating the variable." We want $x$ to be all alone on one side of the scale so we can see what it equals (e.g., $x = 5$).
The Golden Rule of Algebra
Here is the most important rule you will learn this month:
Whatever you do to one side of the equation, you MUST do to the other.
If you add 5 to the left, you must add 5 to the right. If you divide the left by 2, you divide the right by 2. If you don't, the scale tips, and your answer is wrong.
How to Solve: Inverse Operations (The "Undo" Button)
To get $x$ alone, we have to get rid of the numbers crowding it. We do this by doing the opposite.
If you see addition ($+$), you use subtraction ($-$).
If you see subtraction ($-$), you use addition ($+$).
If you see multiplication (like $3x$), you use division ($\div$).
If you see division (like $\frac{x}{4}$), you use multiplication ($\times$).
Example 1:
The problem: $x$ is being annoyed by a "minus 7".
The fix: The opposite of minus 7 is plus 7.
The work: Add 7 to both sides.
$$x - 7 + 7 = 10 + 7$$The result: The $-7$ and $+7$ cancel out (poof!).
$$x = 17$$
Example 2:
The problem: $4x$ means 4 times $x$. $x$ is being multiplied by 4.
The fix: The opposite of multiplication is division.
The work: Divide both sides by 4.
$$\frac{4x}{4} = \frac{20}{4}$$The result:
$$x = 5$$
Boom. That’s it. Keep it balanced, do the opposite, and isolate that variable!
📝 Week 1 Assessment: One-Step Equations
Instructions: Choose the best answer. No calculators needed!
In the equation $x + 5 = 15$, what is the first step to isolate $x$?
A) Add 5 to both sides
B) Subtract 5 from both sides
C) Divide by 5
D) Multiply by 5
Answer: B. Rationale: The inverse of adding 5 is subtracting 5.
Solve for $y$: $y - 8 = 12$
A) 4
B) 18
C) 20
D) 96
Answer: C. Rationale: Add 8 to both sides ($12 + 8 = 20$).
What is the inverse operation of multiplication?
A) Addition
B) Subtraction
C) Division
D) Squaring
Answer: C. Rationale: Division "undoes" multiplication.
Solve for $m$: $3m = 21$
A) 7
B) 18
C) 24
D) 63
Answer: A. Rationale: Divide both sides by 3 ($21 \div 3 = 7$).
In the equation $\frac{x}{4} = 5$, what operation is currently happening to $x$?
A) Addition
B) Subtraction
C) Multiplication
D) Division
Answer: D. Rationale: A fraction bar represents division.
Solve for $k$: $\frac{k}{2} = 10$
A) 5
B) 8
C) 12
D) 20
Answer: D. Rationale: Multiply both sides by 2 ($10 \times 2 = 20$).
Identify the variable in this equation: $15 = 3 + z$
A) 15
B) 3
C) $=$
D) $z$
Answer: D. Rationale: The letter representing the unknown value is the variable.
Solve for $x$: $x + 12 = 4$
A) 8
B) 16
C) -8
D) -16
Answer: C. Rationale: Subtract 12 from 4. ($4 - 12 = -8$).
Which equation represents "a number divided by 3 is 9"?
A) $3x = 9$
B) $\frac{x}{3} = 9$
C) $x - 3 = 9$
D) $x + 3 = 9$
Answer: B. Rationale: Division is shown as a fraction.
Solve for $p$: $-5p = 25$
A) 5
B) -5
C) 20
D) 30
Answer: B. Rationale: Divide 25 by -5. Positive divided by negative is negative.
True or False: $x = 5$ is a solution to $x + 3 = 10$.
A) True
B) False
Answer: B. Rationale: Plug it in: $5 + 3 = 8$, not 10.
Solve for $a$: $a + 2.5 = 5.0$
A) 2.0
B) 3.5
C) 2.5
D) 7.5
Answer: C. Rationale: $5.0 - 2.5 = 2.5$.
What is the goal when solving an algebraic equation?
A) To get the variable equal to zero
B) To isolate the variable on one side
C) To move all numbers to the left
D) To get the biggest number possible
Answer: B. Rationale: We want to find what the variable equals (e.g., $x = \dots$).
Solve for $w$: $w - (-4) = 10$ (Hint: Minus a negative is a plus!)
A) 14
B) 6
C) -6
D) -14
Answer: B. Rationale: $w - (-4)$ becomes $w + 4 = 10$. Subtract 4 to get 6.
If $4x = 0$, what is $x$?
A) 4
B) 0
C) 1
D) Undefined
Answer: B. Rationale: 0 divided by 4 is 0.
Solve for $b$: $b - 100 = 250$
A) 150
B) 250
C) 350
D) 100
Answer: C. Rationale: Add 100 to both sides.
Which step justifies moving from $x + 3 = 8$ to $x = 5$?
A) Addition Property of Equality
B) Subtraction Property of Equality
C) Multiplication Property of Equality
D) Distributive Property
Answer: B. Rationale: You subtracted from both sides.
Solve for $x$: $-x = 5$ (Hint: This is like $-1x = 5$)
A) 5
B) -5
C) 1
D) 0
Answer: B. Rationale: Divide by -1. $5 \div -1 = -5$.
The scale is balanced. The left side has $x$ and two 1-lb weights. The right side has five 1-lb weights. What is $x$?
A) 2
B) 3
C) 5
D) 7
Answer: B. Rationale: Remove 2 weights from both sides ($5 - 2 = 3$).
Solve for $t$: $7t = 49$
A) 6
B) 7
C) 8
D) 9
Answer: B. Rationale: $49 \div 7 = 7$.
Would you like me to proceed to Week 2: "Double Trouble (Two-Step Equations)"?
Here is the breakdown for Week 2, where we level up from simple moves to "combos."
🎓 Week 2: Double Trouble (Two-Step Equations)
🗣️ Lecture: "Socks, Shoes, and the Backwards Walk"
(Tone: 8th Grade Math Enthusiast)
Welcome to Week 2. You conquered the one-step equations. You know how to isolate $x$ when there's just one number bothering it. But what happens when $x$ is surrounded by a crowd?
Look at this equation:
Here, $x$ is being multiplied by 2 AND added by 5. It’s a "Two-Step" equation because we need two moves to solve it. But which move comes first? Do we divide by 2 first, or subtract 5?
The Golden Rule of Week 2: Socks and Shoes
Think about getting dressed in the morning.
First, you put on your Socks ($x$).
Then, you put on your Shoes ($+ 5$).
Now, think about getting undressed at night. You have to do the reverse order:
Take off the Shoes first (undo the $+ 5$).
Take off the Socks last (undo the $x$).
In math, we usually follow PEMDAS (Order of Operations). But when we are solving (or "undressing" the equation), we do PEMDAS in Reverse.
Meet SADMEP
This is PEMDAS spelled backward. This is your map for solving equations.
S / A (Subtraction / Addition): Undo these first. Move the "loose numbers" away from the variable.
D / M (Division / Multiplication): Undo these second. Peel the number stuck directly to the variable.
Let's Solve: $2x + 5 = 15$
Step 1: SADMEP says undo Addition/Subtraction first.
We see a $+ 5$. The opposite is $- 5$.
$$2x + 5 - 5 = 15 - 5$$$$2x = 10$$(See? Now it looks like a Week 1 problem!)
Step 2: SADMEP says undo Division/Multiplication next.
We see $2x$ (multiplication). The opposite is division.
$$\frac{2x}{2} = \frac{10}{2}$$$$x = 5$$
Wait, what about division problems?
Let's try: $\frac{x}{3} - 4 = 2$
Step 1 (Undo Subtraction): Add 4 to both sides.
$$\frac{x}{3} = 6$$Step 2 (Undo Division): Multiply both sides by 3.
$$x = 18$$
Summary: Move the constants (the lonely numbers) first. Handle the coefficient (the number touching $x$) last.
📝 Week 2 Assessment: Two-Step Equations
Instructions: Choose the best answer. Remember: Shoes off, then socks off!
In the equation $3x - 7 = 14$, what is the first step?
A) Divide by 3
B) Subtract 7
C) Add 7
D) Multiply by 3
Answer: C. Rationale: Undo subtraction/addition first. Inverse of $-7$ is $+7$.
Solve for $x$: $2x + 4 = 12$
A) 2
B) 4
C) 6
D) 8
Answer: B. Rationale: Subtract 4 ($2x=8$), then divide by 2 ($x=4$).
Solve for $y$: $5y - 10 = 25$
A) 3
B) 5
C) 7
D) 15
Answer: C. Rationale: Add 10 ($5y=35$), then divide by 5 ($y=7$).
In the equation $\frac{x}{2} + 3 = 8$, what is the first step?
A) Multiply by 2
B) Subtract 3
C) Add 3
D) Divide by 2
Answer: B. Rationale: Always remove the constant (+3) before handling the division.
Solve for $m$: $\frac{m}{4} + 2 = 5$
A) 3
B) 7
C) 12
D) 20
Answer: C. Rationale: Subtract 2 ($m/4 = 3$), then multiply by 4 ($m=12$).
Solve for $k$: $-3k + 8 = 20$
A) -4
B) 4
C) -9
D) 9
Answer: A. Rationale: Subtract 8 ($-3k=12$), divide by -3 ($k=-4$).
What is the correct order of operations when solving for a variable?
A) PEMDAS (Parentheses first)
B) SADMEP (Add/Sub first)
C) Always divide first
D) It doesn't matter
Answer: B. Rationale: We work backwards to isolate the variable.
Solve for $x$: $10 = 2x - 6$
A) 2
B) 8
C) 16
D) 32
Answer: B. Rationale: Add 6 ($16=2x$), divide by 2 ($x=8$).
Identify the constant in $4x + 9 = 21$.
A) 4
B) $x$
C) 9
D) 4x
Answer: C. Rationale: The constant is the number not attached to the variable.
Solve for $b$: $\frac{b}{5} - 1 = 9$
A) 2
B) 40
C) 45
D) 50
Answer: D. Rationale: Add 1 ($\frac{b}{5} = 10$), multiply by 5 ($b=50$).
Solve for $x$: $6x + 5 = 5$
A) 1
B) 5
C) 0
D) Undefined
Answer: C. Rationale: Subtract 5 ($6x=0$), divide by 6 ($x=0$).
What is the second step to solve $4x - 8 = 20$?
A) Add 8
B) Subtract 8
C) Multiply by 4
D) Divide by 4
Answer: D. Rationale: First step is Add 8. Second step is Divide by 4.
Solve for $p$: $-x + 4 = 10$ (Remember: $-x$ is $-1x$)
A) 6
B) -6
C) 14
D) -14
Answer: B. Rationale: Subtract 4 ($-x = 6$), divide by -1 ($x = -6$).
If you have the equation $3x + 2 = 14$, and you divide by 3 first, what happens?
A) You get the right answer faster.
B) You have to divide everything (including the 2) by 3, creating messy fractions.
C) The equation explodes.
D) The 2 disappears.
Answer: B. Rationale: You can do it, but you must divide EVERY term. It makes the math harder ($x + \frac{2}{3} = \frac{14}{3}$).
Solve for $z$: $12 - 2z = 4$
A) 4
B) -4
C) 8
D) -8
Answer: A. Rationale: Subtract 12 ($-2z = -8$), divide by -2 ($z = 4$).
Translate to an equation: "Five more than twice a number is 15."
A) $5x + 2 = 15$
B) $2x - 5 = 15$
C) $2x + 5 = 15$
D) $x + 10 = 15$
Answer: C. Rationale: "Twice a number" is $2x$, "five more" is $+5$.
Solve for $x$: $0.5x + 2 = 4$
A) 1
B) 2
C) 4
D) 10
Answer: C. Rationale: Subtract 2 ($0.5x = 2$). Divide by 0.5 ($x=4$).
Solve for $r$: $15 = \frac{r}{-2} + 5$
A) -5
B) -20
C) 20
D) 10
Answer: B. Rationale: Subtract 5 ($10 = \frac{r}{-2}$), multiply by -2 ($r = -20$).
Spot the error: To solve $2x + 1 = 9$, a student divides by 2 first and gets $x + 1 = 4.5$. Why is this wrong?
A) They didn't divide the 1 by 2.
B) They divided 9 by 2 incorrectly.
C) They should have subtracted 1 first.
D) Both A and C.
Answer: D. Rationale: It's cleaner to subtract first, but if you divide first, you MUST divide the 1 by 2 as well.
Solve for $x$: $-2x - 10 = -20$
A) 5
B) -5
C) 15
D) -15
Answer: A. Rationale: Add 10 ($-2x = -10$), divide by -2 ($x = 5$).
Would you like me to proceed to Week 3: "The Balancing Act (Advanced Mechanics)"?
Here is the breakdown for Week 3. Things are about to get a little messy, but we have the tools to clean it up.
🎓 Week 3: The Balancing Act (Advanced Mechanics)
🗣️ Lecture: "The Mirror World and the Package Deal"
(Tone: 8th Grade Math Enthusiast)
Okay, you are officially in the "Pro" zone now. Week 1 was about balance. Week 2 was about order. Week 3 is about organization.
Up until now, $x$ has always been on the left side, chilling out. But what if you see this?
Wait, there are $x$'s on both sides? It’s like looking in a mirror, but the reflection is slightly different.
Rule #1: Pick a Team (Variables on Both Sides)
You cannot solve for $x$ if it's scattered all over the place. You need to get all the $x$'s on one side and all the regular numbers on the other.
The Strategy: Move the "smaller" $x$ to the "bigger" $x$. It keeps things positive and easy.
The Move: In $3x + 4 = 2x + 10$, the $2x$ is smaller. Let's "kill" it by doing the opposite. Subtract $2x$ from both sides.
$$3x - 2x + 4 = 2x - 2x + 10$$$$x + 4 = 10$$(Now solve it like normal! Subtract 4, and $x = 6$.)
Rule #2: The Package Deal (The Distributive Property)
Sometimes $x$ is hiding inside parentheses, like a package.
You cannot touch the $x$ until you open the package. To open it, you have to multiply the number outside by EVERYTHING inside.
Multiply $2$ by $x$ $\rightarrow$ $2x$
Multiply $2$ by $5$ $\rightarrow$ $10$
New equation: $2x + 10 = 20$
Rule #3: Clean Your Room (Combining Like Terms)
Before you start moving things across the equals sign, look at your own side. Is it messy?
Don't start subtracting 5 yet. Combine the $3x$ and $2x$ first. They are "like terms" (they both have an $x$ backpack).
Much better. Now solve.
Summary Checklist:
Distribute (Open parentheses).
Combine (Clean up each side).
Move Variables (Get all $x$'s to one side).
SADMEP (Solve!).
📝 Week 3 Assessment: Advanced Mechanics
Instructions: Choose the best answer. Don't let the parentheses scare you!
Simplify the expression: $3(x + 4)$
A) $3x + 4$
B) $3x + 12$
C) $x + 12$
D) $7x$
Answer: B. Rationale: Multiply 3 by $x$ AND 3 by 4 (Distributive Property).
In the equation $5x = 3x + 10$, what is the best first step?
A) Divide by 5
B) Subtract $3x$ from both sides
C) Add $3x$ to both sides
D) Divide by 3
Answer: B. Rationale: Move the variable terms to one side. $5x - 3x = 2x$.
Solve for $x$: $4x - x = 12$
A) 3
B) 4
C) 6
D) 12
Answer: B. Rationale: Combine like terms first ($4x - 1x = 3x$). Then $3x = 12$, so $x = 4$.
Simplify: $2(3x - 1)$
A) $6x - 1$
B) $5x - 2$
C) $6x - 2$
D) $6x + 2$
Answer: C. Rationale: $2 \cdot 3x = 6x$ and $2 \cdot -1 = -2$.
Solve for $y$: $2(y + 3) = 14$
A) 4
B) 7
C) 5
D) 2
Answer: A. Rationale: Distribute ($2y + 6 = 14$). Subtract 6 ($2y=8$). Divide by 2 ($y=4$).
What is the first step to solve $3(x - 2) = 21$?
A) Add 2
B) Subtract 3
C) Distribute the 3
D) Just guess
Answer: C. Rationale: Open the package (parentheses) first.
Solve for $m$: $5m = 3m + 12$
A) 3
B) 4
C) 6
D) 12
Answer: C. Rationale: Subtract $3m$ from both sides ($2m = 12$), then divide by 2.
Identify the "Like Terms" in this expression: $4x + 3y - 2x + 7$
A) $4x$ and $3y$
B) $3y$ and $7$
C) $4x$ and $-2x$
D) All of them
Answer: C. Rationale: Like terms must have the exact same variable part.
Solve for $x$: $2x + 3 = x + 8$
A) 2
B) 3
C) 5
D) 11
Answer: C. Rationale: Subtract $x$ from both sides ($x + 3 = 8$). Subtract 3 ($x = 5$).
Solve for $k$: $4(k - 2) = 2(k + 6)$
A) 5
B) 10
C) 15
D) 20
Answer: B. Rationale: Distribute ($4k - 8 = 2k + 12$). Subtract $2k$ ($2k - 8 = 12$). Add 8 ($2k = 20$). Divide by 2 ($k=10$).
Which equation has NO solution?
A) $2x + 1 = 2x + 1$
B) $2x + 1 = 2x + 5$
C) $2x = 0$
D) $x + 1 = 5$
Answer: B. Rationale: If you subtract $2x$, you get $1 = 5$, which is impossible.
Which equation has INFINITE solutions?
A) $3x + 2 = 3x + 2$
B) $3x + 2 = 10$
C) $3x = 0$
D) $x = 2$
Answer: A. Rationale: Both sides are identical. $x$ can be anything.
Solve for $x$: $-2(x + 4) = -10$
A) 1
B) 9
C) -1
D) 5
Answer: A. Rationale: Distribute ($-2x - 8 = -10$). Add 8 ($-2x = -2$). Divide by -2 ($x = 1$).
Simplify: $-(x - 5)$
A) $-x - 5$
B) $-x + 5$
C) $x - 5$
D) $x + 5$
Answer: B. Rationale: Distribute the negative sign (multiply by -1). $-1 \cdot -5$ becomes $+5$.
Solve for $p$: $0.5p + 2 = 0.2p + 8$
A) 10
B) 20
C) 30
D) 2
Answer: B. Rationale: Subtract $0.2p$ ($0.3p + 2 = 8$). Subtract 2 ($0.3p = 6$). Divide by 0.3 ($p = 20$).
Why can't you combine $4x$ and $4$?
A) Because they are both even numbers.
B) Because one has a variable and the other is a constant.
C) You can combine them to get $8x$.
D) You can combine them to get 8.
Answer: B. Rationale: You can't add apples ($x$) to oranges (constants).
Solve for $x$: $\frac{1}{2}(4x + 10) = 11$
A) 3
B) 4
C) 2
D) 5
Answer: A. Rationale: Distribute $\frac{1}{2}$ ($2x + 5 = 11$). Subtract 5 ($2x=6$). Divide by 2 ($x=3$).
Spot the error: $3(x + 2) = 12 \rightarrow 3x + 2 = 12$
A) They added instead of subtracted.
B) They didn't distribute the 3 to the 2.
C) They solved for $x$ incorrectly.
D) There is no error.
Answer: B. Rationale: It should be $3x + 6$. You must multiply the 3 by BOTH parts.
Solve for $z$: $5z - 3z + 2 = 12$
A) 2
B) 3
C) 4
D) 5
Answer: D. Rationale: Combine terms ($2z + 2 = 12$). Subtract 2 ($2z=10$). Divide by 2 ($z=5$).
When solving $3x + 10 = 5x - 4$, why is it better to subtract $3x$ from both sides rather than $5x$?
A) It keeps the coefficient of $x$ positive ($2x$).
B) It keeps the numbers smaller.
C) You are required to by law.
D) It doesn't matter, both are equally easy.
Answer: A. Rationale: Dealing with positive coefficients ($2x$) reduces the chance of making a negative sign error.
Would you like me to proceed to Week 4: "Real World Mode (Applications & Inequalities)"?
Here is the breakdown for Week 4, the final frontier. This is where math stops being just numbers on a page and starts describing real life.
🎓 Week 4: Real World Mode (Applications & Inequalities)
🗣️ Lecture: "The Translator and the Flipping Scale"
(Tone: 8th Grade Math Enthusiast)
Congratulations! You made it to the final week. You can move variables, combine terms, and undo operations. Now, we have to answer the question every math student asks: "When am I ever going to use this?"
Part 1: The Translator (Word Problems)
Math is actually a language. It’s just a very shorthand language. Your job is to be the translator between English and Algebra.
Here is your Translation Dictionary:
"Is" / "Equals" / "Result" $\rightarrow$ $=$
"Sum" / "More than" / "Increased by" $\rightarrow$ $+$
"Difference" / "Less than" / "Decreased by" $\rightarrow$ $-$
"Product" / "Of" / "Times" $\rightarrow$ $\times$
"Quotient" / "Per" / "Ratio" $\rightarrow$ $\div$
Example: "Three more than twice a number is 13."
"Three more than" $\rightarrow$ $+ 3$
"Twice a number" $\rightarrow$ $2x$
"Is 13" $\rightarrow$ $= 13$
Equation: $2x + 3 = 13$ (Solve it: $2x=10$, so $x=5$!)
Part 2: The Flipping Scale (Inequalities)
Sometimes, things aren't exactly equal. Sometimes you have "at least" $20 dollars, or you need to be "taller than" 4 feet to ride the coaster. This is where Inequalities come in.
$x > 5$ (Greater than)
$x < 5$ (Less than)
$x \ge 5$ (Greater than or equal to)
$x \le 5$ (Less than or equal to)
Solving them is 99% the same as equations.
If $x + 3 > 10$, you subtract 3 just like normal.
BUT... There is ONE Danger Zone.
Remember the gym scale? If you have a scale that is tipped to the left, and you multiply both sides by a negative number, the whole universe flips upside down.
The Golden Rule of Inequalities:
If you multiply or divide by a NEGATIVE number, you must FLIP the arrow direction.
Example:
Divide by -2.
FLIP THE SIGN! ($<$ becomes $>$)
$x > -5$
If you don't flip it, you're saying something crazy like "negative 2 is bigger than positive 5." Don't do that.
📝 Week 4 Assessment: Applications & Inequalities
Instructions: Choose the best answer. Watch out for negative numbers—they make the signs flip!
Translate "A number decreased by 5 is 10" into an equation.
A) $5 - x = 10$
B) $x - 5 = 10$
C) $x + 5 = 10$
D) $\frac{x}{5} = 10$
Answer: B. Rationale: "Decreased by" means subtraction. Start with $x$, take away 5.
Which symbol represents "at most"? (e.g., You can spend at most $10).
A) $>$
B) $<$
C) $\ge$
D) $\le$
Answer: D. Rationale: "At most" means the max is 10, so it must be less than or equal to 10.
Solve the inequality: $x + 4 < 12$
A) $x < 8$
B) $x > 8$
C) $x < 16$
D) $x = 8$
Answer: A. Rationale: Subtract 4 from both sides. The sign stays the same.
Solve: $-3x > 12$
A) $x > -4$
B) $x < -4$
C) $x > 4$
D) $x < 4$
Answer: B. Rationale: You divided by -3, so you MUST FLIP the sign from $>$ to $<$.
Translate: "The product of 4 and a number is 20."
A) $4 + x = 20$
B) $\frac{x}{4} = 20$
C) $4x = 20$
D) $x - 4 = 20$
Answer: C. Rationale: "Product" indicates multiplication.
When graphing $x \ge 3$ on a number line, what kind of circle do you use?
A) Open circle
B) Closed (filled) circle
C) A square
D) No circle
Answer: B. Rationale: The "or equal to" line means we include 3, so we fill the circle.
Sam buys 3 tacos and a soda ($2). He spends $11 total. Which equation models this?
A) $3x - 2 = 11$
B) $2x + 3 = 11$
C) $3x + 2 = 11$
D) $x + 5 = 11$
Answer: C. Rationale: 3 tacos ($3x$) plus soda ($2$) equals 11.
Solve for $x$: $2x - 5 \ge 15$
A) $x \ge 5$
B) $x \le 10$
C) $x \ge 10$
D) $x > 20$
Answer: C. Rationale: Add 5 ($2x \ge 20$), divide by 2 ($x \ge 10$). Sign does not flip.
Translate: "Five less than twice a number."
A) $5 - 2x$
B) $2x - 5$
C) $5 < 2x$
D) $2x < 5$
Answer: B. Rationale: "Less than" (as a phrase, not the inequality symbol "is less than") reverses the order. You take 5 away FROM the $2x$.
Which inequality requires flipping the symbol?
A) $4x < 12$
B) $x - 5 > 10$
C) $\frac{x}{-2} < 5$
D) $x + (-2) > 5$
Answer: C. Rationale: You have to multiply by a negative (-2) to isolate x.
You have $50 to spend. A ticket is $10. How many snacks ($5 each) can you buy?
A) $10 + 5x \le 50$
B) $10 + 5x \ge 50$
C) $5x - 10 = 50$
D) $10x + 5 = 50$
Answer: A. Rationale: Ticket + Snacks must be less than or equal to money in pocket ($50).
Solve: $5 - x > 10$
A) $x > -5$
B) $x < -5$
C) $x > 5$
D) $x < 5$
Answer: B. Rationale: Subtract 5 ($-x > 5$). Divide by -1 and FLIP sign ($x < -5$).
What does an open circle on a number line mean?
A) The number is included in the answer.
B) The number is NOT included (boundary point).
C) The answer is zero.
D) You made a mistake.
Answer: B. Rationale: It represents strictly $<$ or $>$.
Translate: "The sum of a number and 6 is at least 15."
A) $x + 6 > 15$
B) $x + 6 < 15$
C) $x + 6 \ge 15$
D) $x + 6 \le 15$
Answer: C. Rationale: "At least" means 15 is the minimum, so it can be equal to or greater.
Solve: $3(x - 1) < 9$
A) $x < 2$
B) $x < 4$
C) $x > 4$
D) $x < 3$
Answer: B. Rationale: Distribute ($3x - 3 < 9$). Add 3 ($3x < 12$). Divide by 3 ($x < 4$).
If $x < 5$ and you multiply by 2, what happens?
A) $2x < 10$
B) $2x > 10$
C) $x < 2.5$
D) The world ends.
Answer: A. Rationale: You multiplied by a POSITIVE number, so the sign stays the same.
Which value is a solution to $2x + 1 > 7$?
A) 0
B) 2
C) 3
D) 4
Answer: D. Rationale: Solve ($2x > 6 \rightarrow x > 3$). 4 is the only option greater than 3.
Translate: "Half of a number is equal to 8."
A) $2x = 8$
B) $\frac{1}{2}x = 8$
C) $x - \frac{1}{2} = 8$
D) $x + \frac{1}{2} = 8$
Answer: B. Rationale: "Half of" means multiply by $0.5$ or divide by 2.
A taxi charges $3 flat rate plus $2 per mile. You can spend $15. What inequality models the miles ($m$) you can travel?
A) $3m + 2 \le 15$
B) $2m + 3 \ge 15$
C) $2m + 3 \le 15$
D) $2m - 3 \le 15$
Answer: C. Rationale: Variable is miles ($2m$), constant is $3. Total must be $\le 15$.
Why do we shade the line on an inequality graph?
A) To make it look cool.
B) Because there is only one answer.
C) Because there are infinite answers (a whole range).
D) Because the pencil slipped.
Answer: C. Rationale: Inequalities represent a range of solutions (e.g., any number bigger than 5).
📘 The Grand Review: Mastering Algebraic Equations
(A Comprehensive Guide for the Student)
Introduction: The Language of the Universe
Over the past four weeks, you have embarked on a journey that is arguably the most important academic step you will take in your school career. You moved from arithmetic—the simple calculation of numbers—to Algebra, the logic of problem-solving. Arithmetic asks, "What is $5 + 5$?" Algebra asks, "If I have the answer 10, how did I get there?"
This review summarizes the four critical pillars of solving algebraic equations. If you can master these four concepts, you are ready not just for high school math, but for physics, chemistry, computer science, and economics. They all speak this language.
Week 1 Recap: The Art of Balance and Isolation
In Week 1, we introduced the fundamental philosophy of Algebra: Equality is Balance. The equals sign ($=$) is not a command to "calculate the answer." It is a statement that two things have the same value.
The Goal: Isolate the variable ($x$). We want to know what the mystery box holds.
The Method: Inverse Operations.
To kill addition, we subtract.
To kill subtraction, we add.
To kill multiplication, we divide.
To kill division, we multiply.
We learned that whatever you do to the left side, you must do to the right. If you add 5 to the left but forget the right, your equation becomes a lie. We started with simple One-Step Equations like $x - 7 = 10$. By adding 7 to both sides, we found $x = 17$. This simple mechanic is the heartbeat of all future math.
Week 2 Recap: Unraveling the Knots (SADMEP)
In Week 2, the problems got tougher. $x$ wasn't just hiding behind a number; it was tied up in knots. We encountered Two-Step Equations like $3x + 4 = 19$. To solve these, we needed a map.
We learned that when you evaluate math, you use PEMDAS. But when you solve equations—when you are trying to find $x$—you use SADMEP. You have to undo the equation in reverse order.
Undo Addition/Subtraction first. Move the friends (constants) away from the variable.
Undo Multiplication/Division second. Separate the variable from its coefficient.
This "Socks then Shoes" analogy is crucial. You can't take your socks off while your shoes are still on. In $3x + 4 = 19$, you must remove the $+4$ before you can touch the $3$. This discipline of order prevents messy fractions and confusion.
Week 3 Recap: Advanced Tactics and Cleaning Up
Week 3 was the messy middle. We introduced equations where $x$ was everywhere—on the left, on the right, and stuck inside parentheses. We learned three major cleanup strategies:
Distributive Property: If you see $3(x + 2)$, you cannot solve yet. You must "distribute" the 3 to unlock the parentheses, turning it into $3x + 6$. This is non-negotiable. You cannot solve what you cannot reach.
Combining Like Terms: Before moving things across the border (the equals sign), clean up your own backyard. If you have $2x + 5x$, turn it into $7x$. It makes life easier.
Variables on Both Sides: When $x$ is on both sides (e.g., $5x = 3x + 10$), we learned to "pick a team." We usually move the smaller variable to the larger one using inverse operations. This consolidates our mystery into a single location, allowing us to use our SADMEP skills.
Week 4 Recap: Real World Translation and The Flip
Finally, in Week 4, we connected the abstract symbols to reality. We learned that math is a language of translation. A "sum" is addition; a "product" is multiplication. We learned to take a story about buying tacos or riding taxis and turn it into a solvable math sentence.
We also introduced Inequalities—the math of limits. We learned that solving $2x + 1 < 9$ is almost identical to solving equations, with one deadly trap: The Negative Flip. When you multiply or divide by a negative number, the relationship reverses. "Less than" becomes "greater than." This rule preserves the logic of the number line.
Conclusion: Why This Matters
Why did we spend four weeks on this? Because algebra is the "gatekeeper." It is the tool that allows you to model the world.
Computer Programmers use variables to store data.
Engineers use equations to ensure bridges don't collapse.
Business Owners use inequalities to calculate profit margins (Profit $>$ Costs).
You now possess the toolkit to solve for the unknown. You know that if you stay balanced, follow the order of operations, and keep your signs straight, there is no problem you cannot unravel.
Keep practicing. Math is not a spectator sport; it is a skill, like basketball or playing the guitar. The more equations you solve, the slower the game will feel, and the more confident you will become. You have the foundation. Now, go solve for $x$.
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