Math Practice — Algebra 1 & Geometry

Math Study Guide · Self-Study Edition

Algebra &
Geometry

Core Problems · Tricky Traps · Quick Memory Keys

Name

Date

∑√π

📐 Algebra 1 10 Essential Problems

Most Common

Solve for $x$:
$3x - 7 = 2x + 5$

"Move NUMBERS to the right, VARIABLES to the left."
ISOLATE x = Get x alone on one side

✏️ Example — Try this first Solve $2x + 3 = x + 9$
Step 1: Subtract $x$ from both sides → $x + 3 = 9$
Step 2: Subtract 3 → $x = 6$ ✓

Now you try: $3x - 7 = 2x + 5$

⚠️ Tricky Trap

Solve for $x$:
$-2(x + 4) = 10$

⚠️ Don't forget: the negative sign distributes to BOTH terms inside!

DISTRIBUTE first → then isolate
$-2(x+4)$ means $-2 \cdot x$ AND $-2 \cdot 4$

✏️ Example $-3(x + 2) = 6$
→ $-3x - 6 = 6$
→ $-3x = 12$
→ $x = -4$ ✓

Core Skill

Simplify the expression:
$4x^2 - 3x + 2x^2 + 5x - 1$

LIKE TERMS = Same variable, same exponent
Group: $x^2$ with $x^2$, $x$ with $x$, numbers with numbers

✏️ Example $3x^2 + x + x^2 - 2x + 4 = 4x^2 - x + 4$
↳ Think of $x^2$ as 🍎 and $x$ as 🍊 — only add same fruits!

⚠️ Tricky Trap

Which of the following is the correct factored form of:
$x^2 + 5x + 6$

⚠️ Students often get the signs wrong! Check: does (+)×(+) give the right middle term?

FACTORING key: Find two numbers that
MULTIPLY to $c$ (last) AND ADD to $b$ (middle)

A.$(x + 1)(x + 6)$

B.$(x + 2)(x + 3)$

C.$(x - 2)(x - 3)$

D.$(x + 5)(x + 1)$

✏️ How to Check For $(x+a)(x+b)$: need $a \times b = 6$ AND $a + b = 5$
Try: $2 \times 3 = 6$ ✓ and $2 + 3 = 5$ ✓

Graph Skill

A line has the equation $y = \dfrac{2}{3}x - 4$.
What is the slope? What is the y-intercept?

y = mx + b
$m$ = slope (how steep 📈) | $b$ = y-intercept (where it crosses y-axis)

✏️ Example $y = -3x + 7$ → slope $= -3$, y-intercept $= 7$
Negative slope = line goes DOWN left to right ↘️

⚠️ Most Missed

Solve the inequality and graph on a number line:
$-4x \geq 12$

⚠️ #1 mistake: Forgetting to FLIP the inequality sign when dividing by a negative!

FLIP the sign when multiplying or dividing by a NEGATIVE
$-4x \geq 12 \Rightarrow x \leq -3$ ← sign flipped!

✏️ Example $-2x \geq 8$
Divide both by $-2$ → flip $\geq$ to $\leq$ → $x \leq -4$
Number line: ●—— (closed circle, arrow left)

[ Draw your number line here ]

Systems

Solve the system of equations:
$\begin{cases} 2x + y = 8 \\ x - y = 1 \end{cases}$

ELIMINATION: Add equations to cancel a variable
SUBSTITUTION: Plug one equation into the other

✏️ Hint: Try Elimination Add both equations: $(2x + y) + (x - y) = 8 + 1$
→ $3x = 9$ → $x = 3$ → plug back in to find $y$

⚠️ Tricky Trap

Evaluate: if $f(x) = x^2 - 3x + 1$, find $f(-2)$

⚠️ Students forget: $(-2)^2 = +4$, NOT $-4$!

f(x) = "plug in x" | Negative squared = POSITIVE
$(-2)^2 = (-2) \times (-2) = +4$ always!

✏️ Example $f(-1)$ for $f(x) = x^2 + 2x$:
→ $(-1)^2 + 2(-1) = 1 - 2 = -1$ ✓

Exponent Rules

Simplify: $\dfrac{x^5 \cdot x^3}{x^4}$

MULTIPLY → ADD exponents | DIVIDE → SUBTRACT
$x^a \cdot x^b = x^{a+b}$ and $\dfrac{x^a}{x^b} = x^{a-b}$

✏️ Example $\dfrac{x^6 \cdot x^2}{x^3} = \dfrac{x^8}{x^3} = x^5$ ✓

⚠️ Word Problem

A school store sells pencils for $\$0.50$ each and notebooks for $\$2.00$ each. Sam spends exactly $\$7.00$ buying a total of 8 items. How many pencils and notebooks did Sam buy?

DEFINE variables first → Write 2 equations → Solve system
Let $p$ = pencils, $n$ = notebooks

✏️ Set-Up Hint Equation 1 (items): $p + n = 8$
Equation 2 (cost): $0.5p + 2n = 7$
→ Now solve! Use substitution or elimination.

— 1 —

Algebra 1 ✦ 10 Problems

📏 Geometry 10 Essential Problems

Angle Basics

Two angles are supplementary. One angle measures $3x + 10$° and the other measures $x + 50$°.
Find the value of $x$ and both angles.

SUPPLEMENTARY = adds up to 180° (straight line)
COMPLEMENTARY = adds up to 90° (right angle)
"S comes before C" → 180° before 90°

✏️ Example Two supplementary angles: $2x + 30$ and $x + 60$
→ $(2x+30) + (x+60) = 180$ → $3x + 90 = 180$ → $x = 30$

Pythagorean Theorem

A right triangle has legs of length $6$ and $8$. Find the length of the hypotenuse.

$a^2 + b^2 = c^2$ (c = hypotenuse, always the longest side)
Common triples to memorize: 3-4-5, 5-12-13, 8-15-17

✏️ Example Legs = 3 and 4 → $3^2 + 4^2 = 9 + 16 = 25$ → $c = \sqrt{25} = 5$ ✓

[ Sketch the right triangle and label a, b, c ]

⚠️ Most Missed

Find the area of a triangle with base $= 10$ cm and height $= 6$ cm.
Why can't you just multiply base × height?

⚠️ Most students forget the ½! Triangle = HALF of a parallelogram.

Area of Triangle $= \dfrac{1}{2} \times b \times h$
Think: 2 triangles → 1 rectangle. So 1 triangle = HALF!

✏️ Contrast Rectangle $b=10, h=6$: Area $= 60$ cm²
Triangle $b=10, h=6$: Area $= \frac{1}{2}(60) = 30$ cm² ← half!

Parallel Lines

Two parallel lines are cut by a transversal. One angle measures $65$°.
Name and find three other angle relationships.

Corresponding = SAME position → EQUAL
Alternate Interior = Z-shape → EQUAL
Co-interior (Same-side) = C-shape → ADD to 180°

✏️ Quick Visual If one angle = 65°:
• Corresponding angle = 65° (same side, same position)
• Alternate interior = 65° (Z-shape, across)
• Co-interior = 180° − 65° = 115° (same side, C-shape)

[ Draw two parallel lines with a transversal and label all 8 angles ]

⚠️ Tricky Trap

Find the circumference AND area of a circle with radius $= 5$ cm.
(Leave your answer in terms of $\pi$)

⚠️ Don't mix up the formulas! One uses $r$, one uses $r^2$.

Circumference $= 2\pi r$ (around the outside, 1D)
Area $= \pi r^2$ (inside space, 2D — so $r$ gets squared!)
Memory: "Cherry Pie → $C = 2\pi r$" "Apple Pie → $A = \pi r^2$"

✏️ Example Radius = 3: $C = 2\pi(3) = 6\pi$ cm | $A = \pi(3)^2 = 9\pi$ cm²

Interior Angles

A polygon has 7 sides (heptagon). What is the sum of its interior angles? If it is regular, what is each interior angle?

Sum of interior angles $= (n - 2) \times 180°$
Each angle of a regular polygon $= \dfrac{(n-2) \times 180°}{n}$

✏️ Example Hexagon ($n=6$): Sum $= (6-2) \times 180 = 720°$
Each angle $= 720 \div 6 = 120°$

⚠️ Most Missed

Two triangles are similar: △ABC ~ △DEF. If $AB = 6$, $BC = 9$, and $DE = 4$, find $EF$.

⚠️ Set up the ratio correctly — match corresponding sides!

Similar triangles = Same shape, different size
Corresponding sides are PROPORTIONAL:
$\dfrac{AB}{DE} = \dfrac{BC}{EF}$

✏️ Example $\dfrac{6}{4} = \dfrac{9}{EF}$ → Cross multiply: $6 \cdot EF = 36$ → $EF = 6$

Volume

Find the volume of a cylinder with radius $= 4$ cm and height $= 10$ cm.
(Leave in terms of $\pi$)

Volume of Cylinder $= \pi r^2 h$
Think: Cylinder = stack of circles → Area of circle × height
$\text{Cone} = \frac{1}{3}\pi r^2 h$ (Cone is 1/3 of cylinder!)

✏️ Example Radius $= 3$, height $= 5$: $V = \pi(3)^2(5) = 45\pi$ cm³

⚠️ Coordinate Geometry

Find the distance between points $A(1, 2)$ and $B(7, 10)$.
Also find the midpoint of segment $AB$.

⚠️ The Distance Formula IS the Pythagorean theorem in disguise!

Distance $= \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
Midpoint $= \left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)$
Midpoint = AVERAGE of x's and AVERAGE of y's

✏️ Example — Midpoint only $A(2,4)$ and $B(8,10)$: Midpoint $= \left(\frac{2+8}{2}, \frac{4+10}{2}\right) = (5, 7)$

⚠️ Challenge

A rectangular prism has length $= 5$ cm, width $= 4$ cm, height $= 3$ cm.
Find (a) the Volume and (b) the Total Surface Area.

⚠️ Surface area counts ALL 6 faces — don't forget top and bottom!

Volume $= l \times w \times h$ (how much fits inside)
Surface Area $= 2(lw + lh + wh)$ (all 6 faces wrapped)
3 pairs of opposite faces → multiply each pair by 2!

✏️ Example — SA Formula Unpacked $l=2, w=3, h=4$:
Front+Back: $2(2\times4)=16$ | Left+Right: $2(3\times4)=24$ | Top+Bottom: $2(2\times3)=12$
Total SA $= 16+24+12 = 52$ cm²

[ Sketch the 3D rectangular prism and label all dimensions ]

— 2 —

Geometry ✦ 10 Problems