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Praxis 5165 โ Overview & Strategy
The Mathematics: Content Knowledge test covers the full secondary curriculum: Number & Quantity, Algebra, Functions, Calculus, Geometry, and Statistics & Probability. About one-quarter of the questions are set in a teaching context โ you evaluate a student's work, choose the best instructional example, or judge whether a method is valid.
Format facts. Selected-response and numeric-entry questions, ~150 minutes. An on-screen graphing calculator is provided, and a formula/notation sheet is on the Help screen.
How to work smart
Decide your approach first, then reach for the calculator. Many items are faster by hand; use the calculator for the last step or two.
Don't round intermediate results. Keep full precision in the calculator; round only the final answer to match the choices.
Re-read what's asked. If your value isn't a choice, you may have answered a different question or rounded too early.
Check the calculator mode (degrees vs. radians) before any trig.
Test tip: For "which student is correct / which example is best" items, the math is usually easy โ the skill is spotting the flawed reasoning. Read every option; the trap answer is often a correct answer reached by invalid steps.
Number & Quantity Essentials
The real numbers include the rationals (any \(\tfrac{a}{b}\) with integers \(a,b\), \(b\neq0\) โ these have terminating or repeating decimals) and the irrationals (non-repeating, non-terminating, like \(\sqrt{2}\) and \(\pi\)).
Closure facts to memorize. rational \(+\) rational \(=\) rational; rational \(\times\) rational \(=\) rational. But rational \(+\) irrational \(=\) irrational, and irrational \(\times\) irrational can be either (e.g. \(\sqrt2\cdot\sqrt2=2\) is rational, \(\sqrt2\cdot\sqrt3=\sqrt6\) is irrational).
Number theory quick hits
Prime factorization is the backbone of GCF/LCM: \(GCF\) = product of shared primes at lowest powers; \(LCM\) = all primes at highest powers.
A number is even/odd by its factor of 2; know divisibility rules (3: digit sum; 9: digit sum; etc.).
Proportional reasoning & units
Most "real world" number items are proportions or percents. Set up a proportion or a unit-conversion chain and let units cancel.
Percent-of-a-whole. If a part is a known percent of an unknown whole, divide: whole \(=\dfrac{\text{part}}{\text{percent as decimal}}\). Example: a \$8.10 price that includes 8% tax came from \(\dfrac{8.10}{1.08}=\$7.50\).
Worked example โ dimensional analysis
Convert \(60\) miles per hour to feet per second (1 mi \(=5280\) ft, 1 hr \(=3600\) s).
Chain the conversions so units cancel. \(\dfrac{60\ \text{mi}}{1\ \text{hr}}\cdot\dfrac{5280\ \text{ft}}{1\ \text{mi}}\cdot\dfrac{1\ \text{hr}}{3600\ \text{s}}\).
Multiply/divide the numbers. \(\dfrac{60\cdot5280}{3600}=88\).
Result. \(60\) mph \(=88\) ft/s.
Test tip: Write the units in your work and cancel them like factors โ if the leftover units match the answer's units, your setup is right.
Exponents, Radicals & Rational Exponents
The rules. \(x^a x^b=x^{a+b}\); \(\dfrac{x^a}{x^b}=x^{a-b}\); \((x^a)^b=x^{ab}\); \((xy)^a=x^a y^a\); \(x^0=1\); \(x^{-a}=\dfrac{1}{x^a}\); and \(x^{m/n}=\sqrt[n]{x^m}=(\sqrt[n]{x})^m\).
Radicals โ rational exponents
A rational exponent's denominator is the root, numerator is the power: \(8^{2/3}=(\sqrt[3]{8})^2=2^2=4\). This lets you simplify radicals with exponent rules.
Worked example
Simplify \(\left(\dfrac{2x^3y^{-2}}{1}\right)^{3}\) with positive exponents.
Raise each factor to the 3rd power. \(2^3=8\), \((x^3)^3=x^9\), \((y^{-2})^3=y^{-6}\).
Fix the negative exponent. \(y^{-6}=\dfrac{1}{y^6}\).
Result. \(\dfrac{8x^9}{y^6}\).
Scientific notation
Write as \(a\times10^{n}\) with \(1\le|a|<10\). Multiply/divide by handling \(a\)'s and powers of 10 separately, then re-normalize.
Test tip: A negative exponent means "reciprocal," never a negative number. And \((\text{something})^{1/2}=\sqrt{\ }\) โ a half-power is a square root.
Rewriting Expressions & Factoring
A big Praxis theme is rewriting an expression to reveal something: factor to find zeros, complete the square to find a vertex/extremum, or change bases.
Factoring toolkit (in order).
GCF โ always pull it out first.
Difference of squares: \(a^2-b^2=(a+b)(a-b)\).
Trinomial \(x^2+bx+c\): two numbers that multiply to \(c\), add to \(b\).
Trinomial \(ax^2+bx+c\): the ac-method (split the middle, then group).
Perfect square: \(a^2\pm2ab+b^2=(a\pm b)^2\).
Grouping for four terms.
Worked example โ ac-method
Factor \(6x^2+x-2\).
Multiply \(a\cdot c\). \(6\cdot(-2)=-12\); find two numbers that multiply to \(-12\), add to \(+1\): \(+4,-3\).
Split the middle term. \(6x^2+4x-3x-2\).
Group and factor. \(2x(3x+2)-1(3x+2)=(3x+2)(2x-1)\).
Test tip: "Factor completely" means keep going until nothing factors โ a common trap answer stops after the GCF, e.g. \(2(x^2-9)\) instead of \(2(x+3)(x-3)\). The Praxis also expects factoring over the complex numbers: \(x^2+y^2=(x+yi)(x-yi)\).
Quadratic Equations & the Discriminant
Quadratic formula. For \(ax^2+bx+c=0\): \[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.\] The discriminant \(D=b^2-4ac\) tells the nature of the roots: \(D>0\) two real; \(D=0\) one (double) real; \(D<0\) two complex.
Completing the square
Any quadratic can be written \((x-p)^2=q\). Take half of the \(x\)-coefficient, square it, and add/subtract it.
Example: \(x^2+8x+c\) is a perfect square when \(c=\left(\tfrac{8}{2}\right)^2=16\), giving \((x+4)^2\).
Worked example โ complex solutions
Solve \(x^2-4x+13=0\).
Compute the discriminant. \(D=(-4)^2-4(1)(13)=16-52=-36<0\Rightarrow\) complex roots.
Apply the formula. \(x=\dfrac{4\pm\sqrt{-36}}{2}=\dfrac{4\pm6i}{2}\).
Simplify. \(x=2\pm3i\).
Test tip: You can read root behavior without solving โ a graph that never crosses the \(x\)-axis has \(D<0\). And "solve any way you like" items often factor quickly; try factoring before the formula.
Functions โ the Core Ideas
A function assigns exactly one output to each input (vertical-line test on a graph). \(f(3)\) means "evaluate at \(x=3\)."
Domain & range. Domain = allowed inputs (exclude values that make a denominator \(0\) or a square root negative). Range = resulting outputs.
Transformations of \(f(x)\)
\(f(x)+k\): shift up \(k\); \(f(x+k)\): shift left \(k\) (inside works "backwards").
\(k\,f(x)\): vertical stretch by \(k\); \(f(kx)\): horizontal compress by \(k\).
Negatives flip: \(-f(x)\) over the \(x\)-axis, \(f(-x)\) over the \(y\)-axis.
Inverses
An inverse \(f^{-1}\) undoes \(f\): swap \(x\) and \(y\), then solve for \(y\). A function has an inverse only if it's one-to-one (horizontal-line test).
Worked example
Find \(f^{-1}(x)\) for \(f(x)=3x-2\).
Write \(y=3x-2\) and swap. \(x=3y-2\).
Solve for \(y\). \(x+2=3y\Rightarrow y=\dfrac{x+2}{3}\).
Test tip: The graph of \(f^{-1}\) is the reflection of \(f\) over the line \(y=x\); exponential and logarithmic functions are inverses of each other.
Complex Numbers
The imaginary unit is \(i=\sqrt{-1}\), so \(i^2=-1\). A complex number is \(a+bi\) (real part \(a\), imaginary part \(b\)).
Powers of \(i\) cycle every 4: \(i,\,-1,\,-i,\,1,\dots\) To simplify \(i^{n}\), divide \(n\) by 4 and use the remainder (\(i^{50}=i^{2}=-1\)). Conjugate: \(\overline{a+bi}=a-bi\). Modulus: \(|a+bi|=\sqrt{a^2+b^2}\).
Arithmetic
Add/subtract: combine real and imaginary parts separately.
Multiply: FOIL, then replace \(i^2=-1\).
Divide: multiply top and bottom by the conjugate of the denominator to make it real.
Worked example โ division
Write \(\dfrac{3+2i}{1-i}\) in standard form.
Multiply by \(\dfrac{1+i}{1+i}\). Numerator: \((3+2i)(1+i)=3+3i+2i+2i^2=1+5i\).
Test tip: \(\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}\) does not hold for negatives โ convert to \(i\) first: \(\sqrt{-8}\cdot\sqrt{-2}=(2i\sqrt2)(i\sqrt2)=-4\), not \(\sqrt{16}=4\).
Linear Equations, Lines & Systems
Forms of a line. Slope-intercept \(y=mx+b\); point-slope \(y-y_1=m(x-x_1)\); standard \(Ax+By=C\). Slope \(m=\dfrac{y_2-y_1}{x_2-x_1}\). Parallel lines share \(m\); perpendicular slopes are negative reciprocals (\(m_1m_2=-1\)).
Intercepts
\(x\)-intercept: set \(y=0\). \(y\)-intercept: set \(x=0\). In a model, the \(y\)-intercept is the starting value and the slope is the rate of change.
Systems โ three methods
Graphing: the solution is the intersection point.
Substitution: solve one equation for a variable, substitute.
Elimination: add/subtract to cancel a variable.
A linearโquadratic system can have 0, 1, or 2 solutions โ set the expressions equal and solve the resulting quadratic.
Worked example โ elimination
Solve \(2x+y=7\) and \(x-y=2\).
Add the equations to cancel \(y\). \(3x=9\Rightarrow x=3\).
Back-substitute. \(3-y=2\Rightarrow y=1\).
Solution. \((3,1)\).
Test tip: Parallel lines (same slope, different intercepts) give no solution; identical lines give infinitely many.
Rational & Radical Equations
These often introduce extraneous solutions โ values that appear during solving but fail in the original equation.
Always check. For radicals, squaring can create false roots. For rationals, any value that makes a denominator \(0\) is excluded.
Worked example โ radical
Solve \(x=\sqrt{\dfrac{2-x}{3}}\).
Square both sides. \(x^2=\dfrac{2-x}{3}\Rightarrow 3x^2+x-2=0\).
Factor. \((3x-2)(x+1)=0\Rightarrow x=\tfrac23\) or \(x=-1\).
Check. The right side is a principal root (\(\ge 0\)), so \(x\ge 0\): reject \(x=-1\); keep \(x=\tfrac23\).
Test tip: If a Praxis item asks you to "justify the reasoning," the key move is stating why an extraneous solution must be discarded โ not just crossing it out.
Exponential & Logarithmic Functions
Exponential: \(f(x)=a\,b^{x}\) โ grows by equal factors over equal intervals (\(b>1\) growth, \(0<b<1\) decay). Log is its inverse: \(y=\log_b x \iff b^{y}=x\).
Test tip: "Linear grows by equal differences; exponential grows by equal factors." An exponential eventually overtakes any polynomial. Rewrite \(A(t)=Pe^{rt}\) situations by reading \(P\) as the initial amount and \(r\) as the continuous rate.
Trigonometry & the Unit Circle
Radians โ degrees: \(180^\circ=\pi\) rad, so multiply by \(\dfrac{\pi}{180}\) or \(\dfrac{180}{\pi}\). On the unit circle a point is \((\cos\theta,\sin\theta)\); \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\).
Reduce any angle by a multiple of \(2\pi\), find its reference angle, then attach the sign for its quadrant (ASTC: All, Sine, Tangent, Cosine positive).
Right-triangle trig & laws
SOH-CAH-TOA for right triangles. For any triangle: Law of Sines \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\); Law of Cosines \(c^2=a^2+b^2-2ab\cos C\). Complementary angles: \(\sin\theta=\cos(90^\circ-\theta)\).
Test tip: Set the calculator to the mode the problem uses (degrees vs. radians) before evaluating. Know the Pythagorean identity \(\sin^2\theta+\cos^2\theta=1\).
Sequences
Arithmetic (common difference \(d\)): \(a_n=a_1+(n-1)d\). Geometric (common ratio \(r\)): \(a_n=a_1\,r^{\,n-1}\). Each also has a recursive form (\(a_n=a_{n-1}+d\) or \(a_n=r\,a_{n-1}\)).
Worked example
A sequence has \(a_1=5\) and \(d=3\). Find \(a_{10}\).
Use the explicit formula. \(a_{10}=5+(10-1)(3)\).
Compute. \(5+27=32\).
Test tip: To test "is it geometric?", check that consecutive ratios are equal; for arithmetic, consecutive differences. The middle term of a geometric triple satisfies \(b^2=ac\).
Limits & Continuity
\(\lim_{x\to a}f(x)=L\) means \(f(x)\) approaches \(L\) as \(x\) nears \(a\) (from both sides). The limit can exist even where \(f(a)\) is undefined.
End-behavior of rationals. Compare degrees: numerator degree < denominator โ limit \(0\) (horizontal asymptote \(y=0\)); equal degrees โ ratio of leading coefficients; numerator larger โ \(\pm\infty\).
Continuity at \(x=a\) โ the 3-part test
\(f(a)\) is defined,
\(\lim_{x\to a}f(x)\) exists (left = right),
\(\lim_{x\to a}f(x)=f(a)\).
Test tip: A one-sided limit that disagrees (a jump) means the two-sided limit doesn't exist. A function can be continuous but not differentiable at a corner (e.g. \(|x|\) at \(0\)).
Derivatives
The derivative \(f'(x)\) is the instantaneous rate of change โ the slope of the tangent line, the limit of secant slopes.
\(f'>0\): increasing; \(f'<0\): decreasing; \(f'=0\): possible max/min (critical point).
\(f''>0\): concave up; \(f''<0\): concave down; sign change of \(f''\) is an inflection point.
Worked example โ tangent slope
Find the slope of \(f(x)=3x^2-7x+2\) at \(x=2\).
Differentiate. \(f'(x)=6x-7\).
Evaluate. \(f'(2)=12-7=5\).
Test tip: A tangent-and-perpendicular item wants the negative reciprocal of \(f'\). "Increasing and concave down" means \(f'>0\) and \(f''<0\) at that \(x\).
Integrals & the Fundamental Theorem
Antiderivative (power rule). \(\displaystyle\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C\ (n\neq-1)\). FTC: \(\displaystyle\int_a^b f(x)\,dx=F(b)-F(a)\), where \(F'=f\). A definite integral gives (signed) area under the curve.
Worked example
Evaluate \(\displaystyle\int_2^{3}(3t+2)\,dt\).
Antiderivative. \(\tfrac32 t^2+2t\).
Apply FTC. At \(3\): \(\tfrac32(9)+6=19.5\); at \(2\): \(\tfrac32(4)+4=10\).
Subtract. \(19.5-10=\tfrac{19}{2}\).
Test tip: Area between two curves is \(\int (\text{top}-\text{bottom})\,dx\). Positionโvelocityโacceleration are linked by derivatives (down) and integrals (up).
Lines, Angles & Triangles
Parallel lines cut by a transversal: corresponding and alternate-interior angles are congruent; same-side interior angles are supplementary. Vertical angles are congruent.
Triangle facts
Interior angles sum to \(180^\circ\); an exterior angle equals the sum of the two remote interior angles.
Triangle inequality: the sum of any two sides exceeds the third.
Special right triangles: \(45\text{-}45\text{-}90\to\) legs \(x\), hyp \(x\sqrt2\); \(30\text{-}60\text{-}90\to x,\ x\sqrt3,\ 2x\).
Pythagorean theorem: \(a^2+b^2=c^2\).
Worked example โ Law of Cosines
Isosceles triangle with a \(120^\circ\) apex and equal sides \(s\); the base opposite is \(12\). Find \(s\).
Law of Cosines. \(12^2=s^2+s^2-2s^2\cos120^\circ=3s^2\).
Solve. \(s^2=48\Rightarrow s=4\sqrt3\).
Test tip: Congruence criteria: SSS, SAS, ASA, AAS (not SSA). Similarity: AA. Congruent parts follow by CPCTC.
Circles, Polygons, Area & Solids
Circle: circumference \(2\pi r\), area \(\pi r^2\). Arc length \(=\dfrac{\theta}{360^\circ}\cdot2\pi r\); sector area \(=\dfrac{\theta}{360^\circ}\cdot\pi r^2\). A central angle equals its arc; an inscribed angle is half its arc.
Solids
Prism/cylinder volume \(=\) (base area)\(\times\) height. Cone/pyramid \(=\tfrac13\) of that.
Sphere: \(V=\tfrac43\pi r^3\), surface area \(4\pi r^2\).
A cylinder and a cone with the same radius and height satisfy \(V_{\text{cyl}}=3V_{\text{cone}}\).
Test tip: Scaling a length by \(k\) multiplies area by \(k^2\) and volume by \(k^3\). Break irregular figures into triangles/rectangles to find area.
Coordinate Geometry & Transformations
Between two points: distance \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\); midpoint \(\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)\). Circle: \((x-h)^2+(y-k)^2=r^2\) has center \((h,k)\), radius \(r\).
Transformations
Rigid motions (translations, rotations, reflections) preserve distance and angle โ the image is congruent.
Dilations preserve angle but scale distance โ the image is similar.
\(90^\circ\) rotation about the origin: \((x,y)\to(-y,x)\) counterclockwise, \((y,-x)\) clockwise.
Test tip: To show two figures are congruent, exhibit a sequence of rigid motions mapping one onto the other; for similar, allow a dilation too. Perpendicular slopes multiply to \(-1\).
Summarizing Data
Center: mean (average) vs. median (middle of ordered data; resistant to outliers). Spread: range, interquartile range \(\text{IQR}=Q_3-Q_1\), and standard deviation.
Displays & the normal curve
Boxplots show the five-number summary (min, \(Q_1\), median, \(Q_3\), max); the box spans the middle \(50\%\).
Empirical (68โ95โ99.7) rule for a normal distribution: about \(68\%\) within \(1\) SD, \(95\%\) within \(2\), \(99.7\%\) within \(3\).
Worked example
July highs are \(N(\mu=105,\sigma=5)\). What percent fall between \(95^\circ\) and \(100^\circ\)?
z-scores. \(95\to z=-2\), \(100\to z=-1\).
Use the rule. Between \(1\) and \(2\) SD below the mean: \(\dfrac{95\%-68\%}{2}=13.5\%\).
Test tip: Choose statistics that fit the shape โ skewed data โ median & IQR; roughly symmetric โ mean & SD. Adding a value below the mean must change the mean, but may leave the median unchanged.
Probability & Two-Way Tables
Basics. \(P(A)=\dfrac{\text{favorable}}{\text{total}}\). Compound: \(P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)\). Conditional: \(P(A\mid B)=\dfrac{P(A\text{ and }B)}{P(B)}\). Events are independent when \(P(A\mid B)=P(A)\).
Two-way tables
Read joint (a single cell), marginal (a row/column total), and conditional (restrict to one row/column, then divide) frequencies. Expected value \(=\sum(\text{value}\times\text{probability})\).
Worked example โ conditional
Of \(260\) public-transit riders, \(120\) don't work at the mall. Find \(P(\text{not work}\mid\text{public})\).
Restrict to the public-transit row. Total \(260\).
Divide. \(\dfrac{120}{260}=\dfrac{6}{13}\).
Test tip: "At least one" is easiest via the complement: \(P(\text{at least one})=1-P(\text{none})\).
Scatterplots, Regression & Correlation
Line of best fit \(y=mx+b\): the slope \(m\) is the predicted change in \(y\) per one-unit increase in \(x\); \(b\) is the predicted value at \(x=0\). The correlation coefficient \(r\) (from \(-1\) to \(1\)) measures strength and direction of a linear relationship.
Reading a scatterplot
Direction: positive (rises) or negative (falls); form: linear or not; strength: how tightly points hug the line.
Residual \(=\) observed \(-\) predicted; a patternless residual plot supports a linear fit.
Test tip:Correlation is not causation โ a strong \(r\) doesn't prove one variable causes the other. Tight downward band โ strong negative correlation.
The Real Number System
Every number you use in Algebra 2 is a real number, and each real number is either rational or irrational. The real numbers are built from nested sets: the counting numbers sit inside the whole numbers, inside the integers, inside the rationals.
Each set sits inside the next; the irrationals fill out the rest of the reals.
๐กKey Idea
Sets of the Real Numbers
Natural (counting)\(1,2,3,4,\dots\)
Whole\(0,1,2,3,\dots\) โ the naturals plus \(0\)
Integers\(\dots,-2,-1,0,1,2,\dots\) โ the wholes and their opposites
Rationalany \(\dfrac{p}{q}\), integers \(p,q\ (q\neq0)\) โ decimals that terminate or repeat
Irrationaldecimals that never end and never repeat \((\pi,\ \sqrt2,\ \sqrt{13})\)
Realevery rational or irrational number
Because the sets are nested, a number belongs to every set that contains it โ so \(-8\) is an integer, and therefore also rational and real (but not whole or natural).
Example 1Classifying Numbers
Name every set each number belongs to.
\(-8\)Integer โ also Rational, Real
\(\sqrt{13}\)13 isn't a perfect square โ Irrational, Real
\(\dfrac{2}{3}\)A ratio of integers โ Rational, Real
\(27\)A counting number โ Natural, Whole, Integer, Rational, Real
Example 2Simplify First โ Hidden Values
Some numbers look irrational (or undefined) until you simplify.
\(\sqrt{225}=15\)Perfect square โ Natural, Whole, Integer, Rational, Real
\(\sqrt{\tfrac{4}{49}}=\tfrac27\)Perfect-square fraction โ Rational, Real
\(4.5\overline{6}\)Repeating decimal โ Rational, Real
\(\dfrac{-43}{0}\)Division by zero is undefined โ not a real number
On the real number line, every real number is a single point. When you graph an inequality, an open circle means the endpoint is not included (\(<,\,>\)); a closed circle means it is included (\(\le,\,\ge\)); shade in the direction that makes the inequality true.
Test tip:Simplify before you classify. \(\sqrt{225}\) and \(\sqrt{4/49}\) are rational once simplified. A repeating decimal is rational; a decimal that never ends and never repeats (like \(3.050050005\ldots\)) is irrational. Division by \(0\) is undefined โ it isn't in any set.
A property is a rule that is always true. The real-number properties are the moves you are allowed to make in algebra โ they let you rewrite an expression as an equivalent expression, one that has the same value for every input. Because the reals are closed under addition and multiplication, \(a+b\) and \(a\cdot b\) are always real numbers too.
๐กKey Idea
The Real Number Properties
For all real numbers \(a\), \(b\), and \(c\):
Commutative Properties\(a+b=b+a\) and \(a\cdot b=b\cdot a\)
Associative Properties\(a+(b+c)=(a+b)+c\) and \(a(bc)=(ab)c\)
Distributive Property\(a(b\pm c)=ab\pm ac\) and \(\dfrac{a\pm b}{c}=\dfrac{a}{c}\pm\dfrac{b}{c}\)
Identity Properties\(a+0=a\) and \(a\cdot 1=a\)
Inverse Properties\(a+(-a)=0\) and \(a\cdot\dfrac{1}{a}=1,\ a\neq 0\)
Careful: subtraction and division are not commutative or associative โ for example \(5-2\neq 2-5\). The distributive property is the bridge between multiplication/division and addition/subtraction โ it is exactly what lets you expand and, run in reverse, factor.
Example 1Writing an Equivalent Expression
Simplify \(3(x+4)+2x\). Justify each step.
\(3(x+4)+2x\)Write the expression.
\(=3x+12+2x\)Distributive Property
\(=3x+2x+12\)Commutative Property of Addition
\(=(3x+2x)+12\)Associative Property of Addition
\(=5x+12\)Combine like terms.
Check
Let \(x=1\): \(3(1+4)+2(1)=17\) and \(5(1)+12=17\). โ
Test tip: When asked "which property justifies this step?" โ the order changed โ Commutative; the parentheses/grouping moved โ Associative; a factor was multiplied across a sum โ Distributive. Don't confuse the first two: same numbers, reordered = commutative; same order, regrouped = associative.
Before we solve anything, we need a shared vocabulary. Everything in algebra is built from variables, terms, and expressions โ and one of the most important ideas we can ask about an expression is where it equals zero.
๐กKey Idea
Starting Definitions
Variablea quantity that is unknown or can change; usually written as a letter or symbol (like \(x\)).
Terma single number, or a product/quotient of numbers and variables โ pieces joined only by \(\times\) or \(\div\) (e.g. \(5x^2\)).
Expressiona combination of terms joined by \(+\) and \(-\).
So the terms of an expression are the chunks separated by \(+\) and \(-\) signs. In \(5x^2-2x+10\), the terms are \(5x^2\), \(-2x\), and \(10\) โ three terms.
Example 1Count Terms & Evaluate
For \(5x^2-2x+10\): count the terms, then evaluate at \(x=-2\).
\(5x^2,\ -2x,\ 10\)Chunks split by \(+/-\) ⇒ 3 terms
Zeroan input value that makes the whole expression equal \(0\). It's called a zero because plugging it in gives you zero.
Example 2Testing a Zero
Show that \(x=4\) is a zero of \(x^2-2x-8\).
\((4)^2-2(4)-8\)Substitute \(x=4\)
\(16-8-8=0\)Result is \(0\) ⇒ \(x=4\) is a zero \(\checkmark\)
A quadratic expression (degree \(2\)) usually has two zeros. You can find them by factoring and setting each factor to \(0\).
Example 3Two Zeros of a Quadratic
Find the zeros of \(x^2-64\).
\(x^2-64=(x-8)(x+8)\)Difference of squares
\(x-8=0\ \Rightarrow\ x=8\)Set the first factor to \(0\)
\(x+8=0\ \Rightarrow\ x=-8\)Set the second factor to \(0\)
๐กKey Idea
Rational Expressions
Equals zeroa fraction is \(0\) exactly when its numerator is \(0\) (and the denominator isn't).
Undefineda fraction is undefined when its denominator is \(0\) โ you can't divide by zero.
Example 4A Rational Expression
Analyze \(\dfrac{x-10}{x+2}\).
at \(x=0:\ \dfrac{0-10}{0+2}=\dfrac{-10}{2}=-5\)Just substitute
at \(x=10:\ \dfrac{10-10}{10+2}=\dfrac{0}{12}=0\)Numerator \(0\) ⇒ \(x=10\) is the zero
\(x=10\) is the only zero\(x-10=0\) has just one solution
at \(x=-2:\ \dfrac{-12}{0}\) is undefinedDenominator \(0\) ⇒ division by zero
Test tip: To find where a fraction is zero, set the top to \(0\). To find where it's undefined, set the bottom to \(0\). A degree-\(2\) expression has up to two zeros โ factor to find them.
An inequality compares two expressions with \(<\), \(>\), \(\le\), or \(\ge\). Its solution isn't a single number โ it's a whole range of values. You solve it almost exactly like an equation, with one crucial twist.
๐กKey Idea
Solving Inequalities โ the Flip Rule
Same movesAdd, subtract, multiply, or divide both sides โ just like an equation.
The flipWhen you multiply or divide both sides by a negative number, flip the inequality sign (\(<\ \leftrightarrow\ >\), \(\le\ \leftrightarrow\ \ge\)).
Why? Multiplying by \(-1\) reverses order on the number line: \(2<5\), but \(-2>-5\).
Example 1Dividing by a Negative
Solve \(-3x+5<20\).
\(-3x<15\)Subtract \(5\) โ no flip (only \(\pm\))
\(\dfrac{-3x}{-3}>\dfrac{15}{-3}\)Divide by \(-3\) ⇒ flip \(<\) to \(>\)
\(x>-5\)Solution set
Example 2Variables on Both Sides
Solve \(2(x-4)\ge 3x+1\).
\(2x-8\ge 3x+1\)Distribute the \(2\)
\(-x-8\ge 1\)Subtract \(3x\)
\(-x\ge 9\)Add \(8\)
\(x\le -9\)Divide by \(-1\) ⇒ flip \(\ge\) to \(\le\)
๐กKey Idea
Interval Notation
( ) parenthesisvalue is excluded โ use with \(<\) and \(>\) (strict).
[ ] bracketvalue is included โ use with \(\le\) and \(\ge\).
\(\infty\)infinity is never reached, so it always gets a parenthesis: \((\ \text{or}\ )\).
Read left-to-right, smallest to largest:
\(x>3\)\((3,\ \infty)\)
\(x\le -2\)\((-\infty,\ -2\,]\)
\(-1\le x<4\)\([-1,\ 4)\)
๐กKey Idea
Graphing on a Number Line
Open circle โendpoint not included โ use for \(<\) and \(>\).
Closed circle โendpoint is included โ use for \(\le\) and \(\ge\).
Arrow / shadingpoints toward every value in the solution (right for larger, left for smaller).
Try it yourself. Pick the operators from the menus โ or just type an inequality (including and/or compounds) โ and it graphs itself, with the inequality and interval notation shown automatically underneath.
Test tip: The flip only happens when you multiply or divide by a negative โ adding or subtracting a negative never flips the sign. And on a graph, the circle style must match the symbol: hollow for \(<,>\); filled for \(\le,\ge\).
A system of equations is two (or more) equations that share the same variables. Its solution is the ordered pair \((x, y)\) that makes every equation true at once โ graphically, the point where the lines cross. You can find it algebraically two ways: substitution and elimination.
๐กKey Idea
What a Solution Means
One solutionthe lines cross once โ a single \((x,y)\).
No solutionthe lines are parallel โ solving gives a false statement like \(1 = -3\).
Infinitely manythe equations are the same line โ solving gives a true statement like \(4 = 4\).
๐กKey Idea
Method 1 โ Substitution
Step 1Solve one equation for one variable (easiest when a coefficient is \(\pm 1\)).
Step 2Substitute that expression into the other equation and solve.
Step 3Back-substitute to find the second variable.
\(4x + 2y = 8\) is \(2 \times (2x + y = 4)\)The second equation is just the first, doubled
Eliminating gives \(0 = 0\)A true statement โ same line
infinitely many solutionsEvery point on the line works
Which method? Use substitution when a variable is already isolated or easy to isolate (a \(\pm 1\) coefficient). Use elimination when the equations are in \(ax + by = c\) form and coefficients line up (or can be matched by multiplying). Both give the same answer โ pick the faster path.