20 carefully selected problems. Each one is a trap — the most-missed questions with memory-first keys.
Solve $3x - 5 = x + 7$
Step 1: $3x - x = 7 + 5$ → $2x = 12$ → $x = 6$
A student solved $5x - 3 = 2x + 9$ and got $x = 4$.
Is this correct? What is the actual solution?
Twice a number decreased by 4 is 10.
Let $x$ = the number → $2x - 4 = 10$ → $x = 7$
Sarah is 3 years older than twice her brother's age. If Sarah is 19, how old is her brother?
$-2x > 8$ → divide by $-2$ → flip! → $x < -4$
Wrong answer: $x > -4$ ← most students write this
Solve $-3x + 6 \geq 15$. Which graph shows the correct solution on a number line?
Line through $(0, 3)$ and $(2, 7)$:
Slope $m = \frac{7-3}{2-0} = 2$ → equation: $y = 2x + 3$
A line passes through the points $(1, 5)$ and $(3, 11)$.
What is the equation of this line?
$y = 2x + 1$ and $y = -x + 7$
→ $2x+1 = -x+7$ → $3x = 6$ → $x=2, y=5$
Tickets cost $\$8$ for adults and $\$5$ for children. A total of 12 tickets were sold for $\$81$. How many adult tickets were sold?
Factor $x^2 + 5x + 6$: find two numbers → $\times 6$, $+5$ → $2$ and $3$
Answer: $(x+2)(x+3)$
Factor completely: $x^2 - x - 12$
$\frac{x^5 \cdot x^2}{x^3} = x^{5+2-3} = x^4$
Simplify: $\dfrac{(2x^3)^2 \cdot x}{4x^5}$
$f(x) = 2x - 1$: to find $f(3)$, plug in $x = 3$
$f(3) = 2(3) - 1 = 5$
If $f(x) = x^2 - 3x + 2$, what is the value of $f(-1)$?
A shirt costs \$40 and is 20% off.
Discount = $40 \times 0.20 = \$8$ → Sale price = $\$32$
A store increases its price by 15%, then offers a 15% discount on the new price. Is the final price the same as the original?
$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for $ax^2 + bx + c = 0$
A ball is thrown upward. Its height is $h = -4t^2 + 8t + 5$, where $h$ is in meters and $t$ is in seconds.
At what time does the ball hit the ground? (i.e., $h = 0$)
Legs 6 and 8 → $6^2 + 8^2 = 36 + 64 = 100$ → $c = 10$
A ladder 13 ft long leans against a wall. The base of the ladder is 5 ft from the wall.
How high up the wall does the ladder reach?
Vertical angles: equal · Supplementary: sum = 180° · Complementary: sum = 90°
Two parallel lines are cut by a transversal. One co-interior (same-side interior) angle measures $(3x + 20)°$ and the other measures $(2x + 40)°$.
Find the value of $x$.
SSA (two sides + non-included angle) does NOT always prove congruence — this is the "ambiguous case"!
Two triangles share a common side. You know two sides and the angle between those two sides in each triangle are equal.
Which congruence postulate applies?
Similar triangles with sides 3, 4, 5 and 6, ?, 10:
Scale factor = 2 → missing side = $4 \times 2 = 8$
A 6-foot person casts a 4-foot shadow. At the same time, a tree casts a 22-foot shadow.
How tall is the tree?
Trapezoid with bases 6 and 10, height 4:
$A = \frac{1}{2}(b_1 + b_2) \cdot h = \frac{1}{2}(16)(4) = 32$ sq units
A composite figure is made of a rectangle (8 × 5) with a semicircle on top of the 8-unit side.
What is the total area? (Use $\pi \approx 3.14$)
If a central angle is 80°, the arc = 80°, but the inscribed angle on the same arc = 40°
An inscribed angle intercepts an arc of 110°.
What is the measure of the inscribed angle?
Cylinder: $V = \pi r^2 h$ · Cone: $V = \frac{1}{3}\pi r^2 h$ · Sphere: $V = \frac{4}{3}\pi r^3$
An ice cream cone has a radius of 3 cm and a height of 9 cm. A scoop of ice cream (sphere) sits on top with radius 3 cm.
What is the total volume? (Use $\pi \approx 3.14$)
Distance: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
Midpoint: $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$
Point $M(3, -1)$ is the midpoint of segment $\overline{AB}$. If $A = (1, 5)$, what are the coordinates of $B$?
90° CCW: $(x,y) \to (-y,x)$ · 180°: $(x,y) \to (-x,-y)$ · Reflect over x-axis: $(x,y) \to (x,-y)$
Triangle $PQR$ has vertices at $P(2, 3)$, $Q(4, 3)$, $R(3, 6)$.
After a 180° rotation about the origin, what are the new coordinates of $P$?
45-45-90: $x : x : x\sqrt{2}$ · 30-60-90: $x : x\sqrt{3} : 2x$
Trap: students often forget to multiply by $\sqrt{3}$, not $\sqrt{2}$
In a 30-60-90 triangle, the hypotenuse is 14 cm.
What is the length of the longer leg?