Pre-Algebra

Lesson 1: Integers and the Number Line

The integers are the whole numbers together with their opposites: …, −3, −2, −1, 0, 1, 2, 3, …. Picture them as evenly spaced points on the number line, with 0 at the center, positive numbers stretching to the right, and negative numbers to the left. Integers show up everywhere: a temperature of −5°C, a bank overdraft of −$40, or a football loss of 8 yards (−8).

Every integer has an opposite: the number the same distance from 0 on the other side. The opposite of 7 is −7, and the opposite of −4 is 4. The absolute value of a number, written with bars like |−9|, is its distance from 0 — always a non-negative number. So |−9| = 9 and |6| = 6. Comparing integers is simple on the number line: any number to the right is greater. That means −2 > −9 (because −2 sits to the right of −9), even though 9 looks “bigger.”

Adding integers is movement on the number line. Start at the first number, then move right for a positive addend and left for a negative one. Subtracting an integer is the same as adding its opposite: 5 − (−3) = 5 + 3 = 8. A quick rule of thumb: if the signs match, add the magnitudes and keep the sign; if the signs differ, subtract the smaller magnitude from the larger and keep the sign of the “winner.” So −14 + 6 = −8 (14 wins, so the answer is negative), while 14 + (−6) = 8.

Try it yourself: Order these from least to greatest: 7, −3, −10, 2, 0. Then evaluate (−8) + 13 and |−12| − |4|.

Lesson 2: Exponents and Square Roots

An exponent tells you how many times to multiply a base by itself. In 4³, the base is 4 and the exponent is 3, so 4³ = 4 × 4 × 4 = 64. Exponents are shorthand for repeated multiplication, and they show up everywhere: the area of a square with side s is s², the volume of a cube is s³, and computer storage grows in powers of 2. A few rules make exponents manageable: any nonzero number to the power 0 equals 1 (5⁰ = 1); a negative exponent means a reciprocal (2⁻³ = 1/8); and when you multiply same-base powers you add exponents (3² × 3⁰ = 3⁶).

A square root undoes squaring: √49 = 7 because 7² = 49. The square roots of perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...) are whole numbers and worth memorizing. Other roots, like √2, are irrational — their decimals never end or repeat — but we can estimate them: √20 is between √16 = 4 and √25 = 5, closer to 4.5 since 4.5² = 20.25. Square roots also appear in geometry: the diagonal of a 3-by-4 rectangle is √(3² + 4²) = √25 = 5.

Watch out for two classic traps. First, (−3)² = 9, but −3² = −9: the exponent binds tighter than the minus sign unless parentheses say otherwise. Second, √(a + b) is NOT √a + √b: √(9 + 16) = √25 = 5, while √9 + √16 = 3 + 4 = 7. Roots do not distribute over addition.

Try it yourself: Evaluate: (a) 3⁰ − √64, (b) (−2)³ + 2⁰, (c) estimate √50 to the nearest tenth. Answers: (a) 81 − 8 = 73; (b) −8 + 16 = 8; (c) about 7.1, since 7.1² = 50.41.

Lesson 3: Multiplying, Dividing, and Order of Operations

Multiplying and dividing integers follows one clean sign rule: same signs → positive result; different signs → negative result. So (−4)(−7) = 28, (12)(−3) = −36, (−20) ÷ (−5) = 4, and 45 ÷ (−9) = −5. For a chain of multiplications, just count the negative factors: an even number of negatives gives a positive product, an odd number gives a negative product. (−2)(−3)(−1) = −6 because there are three negatives.

An exponent is shorthand for repeated multiplication: 4³ = 4 × 4 × 4 = 64. The base is the number being multiplied; the exponent counts the factors. Watch the sign trap: (−3)² = (−3)(−3) = 9, but −3² = −(3 × 3) = −9, because the exponent applies before the negative sign. Parentheses change everything.

When an expression mixes operations, follow the order of operations: parentheses first, then exponents, then multiplication and division (left to right), then addition and subtraction (left to right) — often remembered as PEMDAS. In 3 + 4 × 2², do the exponent first (2² = 4), then the multiplication (4 × 4 = 16), then the addition: 19. If you added first you'd get 28 — a classic mistake. The order isn't arbitrary: it's the universal agreement that lets everyone read the same expression and get the same answer.

Try it yourself: Evaluate: (a) (−6)(4) ÷ (−3), (b) 2 + 3² × 2, (c) (−5 + 2)² − 10 ÷ 2.

Lesson 4: Fractions, Decimals, and Percents

Fractions, decimals, and percents are three languages for the same idea: a part of a whole. The fraction 3/4, the decimal 0.75, and 75% all name the same quantity. Being fluent in all three — converting instantly between them — is one of the most practical skills in mathematics. To convert a fraction to a decimal, divide numerator by denominator: 5/8 = 0.625. To convert a decimal to a percent, multiply by 100 (move the decimal point two places right): 0.625 = 62.5%. To go from percent to decimal, divide by 100: 62.5% = 0.625.

Adding and subtracting fractions requires a common denominator. For 2/3 + 5/6, the least common denominator is 6: rewrite 2/3 as 4/6, then add to get 9/6 = 3/2 = 1½. Multiplying fractions is easier: multiply numerators, multiply denominators, then simplify — 3/4 × 2/9 = 6/36 = 1/6. Dividing fractions means multiplying by the reciprocal: 5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4. Always simplify at the end by dividing numerator and denominator by their greatest common factor.

Percents are really just hundredths: 40% means 40/100. That makes percent problems into simple multiplication: 30% of 90 = 0.30 × 90 = 27. For percent change, compare the change to the original: a price rising from $40 to $46 increased by $6, so the percent increase is 6/40 = 0.15 = 15%. Tips, discounts, sales tax, and interest are all percent applications — a 20% tip on a $35 bill is 0.20 × 35 = $7, and a 25% discount takes $80 down to $80 − $20 = $60.

Try it yourself: Compute (a) 5/6 − 1/4, (b) 3/5 ÷ 9/10, (c) 15% of 240, (d) write 0.08 as a percent and 7/20 as a decimal.

Lesson 5: Percent Applications — Tax, Tip, and Discount

Percents become powerful when you use them to make money decisions. A discount subtracts a percent of the price: a $60 jacket at 25% off costs 60 − 0.25(60) = 60 − 15 = $45. A faster route is to multiply by the complement: paying 75% means 0.75 × 60 = $45 in one step. Sales tax adds a percent: at 6% tax, a $40 game costs 40 + 0.06(40) = $42.40, or 1.06 × 40 in one step. A tip works like tax: a 20% tip on a $35 meal is 0.20 × 35 = $7, for a $42 total.

Percent change compares a new value to an old one: percent change = (new − old) ÷ old × 100%. If a phone plan drops from $50 to $40, that is (40 − 50) ÷ 50 = −20%, a 20% decrease. If your savings grow from $200 to $230, that is a 15% increase. The denominator is always the original value, not the new one — this is the most common percent mistake. Simple interest is also a percent application: interest = principal × rate × time. Put $500 in an account at 4% annual simple interest for 3 years and you earn 500 × 0.04 × 3 = $60.

Stacked percents do not simply add. A store that marks a $100 item down 30% and then takes an extra 20% off does NOT give 50% off: after the first discount the price is $70, and 20% off $70 is $14, leaving $56 — a 44% total discount. Always apply percents one at a time, to the current price.

Try it yourself: (a) A $25 meal with a 20% tip and 8% tax on the meal — total? (b) A bike's price rises from $120 to $150: percent increase? Answers: (a) Tax: 25 × 1.08 = $27; tip: 0.20 × 25 = $5; total $32. (b) (150 − 120) ÷ 120 = 25% increase.

Key vocabulary: Discount — a percent subtracted from a price; sales tax — a percent added to a purchase; percent change — (new − old) ÷ old, as a percent; simple interest — principal × rate × time.

Lesson 6: Ratios, Rates, and Proportions

A ratio compares two quantities with the same units, like 3 cups of flour to 2 cups of sugar (3:2). A rate compares quantities with different units, like 55 miles per hour or $4.50 per pound. A unit rate scales the comparison to one unit — 120 miles in 3 hours is a unit rate of 40 miles per hour — and unit rates are the secret weapon of smart shopping: the 12-ounce jar at $3.60 costs $0.30/oz, while the 20-ounce jar at $5.50 costs $0.275/oz, so the bigger jar is the better deal.

A proportion is an equation stating that two ratios are equal: 3/4 = x/12. Proportions solve “scaling” problems — maps, models, recipes, similar figures. The workhorse technique is cross-multiplication: if a/b = c/d, then a·d = b·c. For 3/4 = x/12, cross-multiplying gives 4x = 36, so x = 9. This works because multiplying both sides by the common denominator b·d clears the fractions.

Watch out for the additive trap: a ratio relationship is multiplicative, not additive. If a 2-inch photo is scaled so its width becomes 6 inches, the scale factor is 3, and a 5-inch height becomes 15 inches — not 9. Always ask: “what multiplies the old value to give the new one?” That multiplier is your scale factor, and it applies to every length in the figure.

Try it yourself: Solve: (a) 5/8 = x/40, (b) a map scale is 1 cm : 25 km; two cities are 9 cm apart on the map — what is the real distance? (c) Which is cheaper per ounce: 16 oz for $4.80 or 24 oz for $6.72?

Lesson 7: Expressions and One-Step Equations

An algebraic expression like 3x + 5 combines numbers, variables, and operations. The 3 is the coefficient of the variable x, and 5 is the constant term. Expressions don't have an equals sign — they represent a quantity that changes as x changes. To evaluate an expression, substitute a value for the variable and compute: 3x + 5 at x = 4 is 3(4) + 5 = 17. Like terms have identical variable parts and can be combined: 4x + 7x = 11x, but 4x + 7y cannot be simplified further.

An equation states that two expressions are equal, like x + 9 = 21. Solving it means finding the value(s) of the variable that make the statement true. For one-step equations, undo the single operation with its inverse: addition undoes subtraction, multiplication undoes division. Whatever you do to one side, you must do to the other — the equation is a balance, and keeping it balanced is what keeps your solution valid.

The four one-step patterns: x + a = b → x = b − a; x − a = b → x = b + a; a·x = b → x = b ÷ a; x ÷ a = b → x = b · a. Notice x ÷ 4 = 12 solves to x = 48 (multiply both sides by 4), while 4/x = 2 solves to x = 2 (multiply both sides by x, then divide by 2). Always check by substituting back into the original equation.

Try it yourself: Solve and check: (a) y − 15 = 7, (b) x/6 = 9, (c) −5n = 55, (d) write an equation for: “a number doubled, then increased by 11, equals 31” — and solve it.

Lesson 8: Two-Step Equations

Many real equations need two moves to isolate the variable. The strategy is to undo in reverse order of operations: first undo the addition or subtraction, then undo the multiplication or division. For 3x + 7 = 22, subtract 7 from both sides to get 3x = 15, then divide by 3 to get x = 5. Think of it as peeling an onion from the outside in: the +7 is the outermost layer, so it comes off first. Always check by substituting back: 3(5) + 7 = 22. Balanced.

Equations with the variable on both sides need one extra idea: gather the variable terms on one side first. For 5x − 8 = 2x + 7, subtract 2x from both sides (3x − 8 = 7), then add 8 (3x = 15), then divide (x = 5). Equations with parentheses need distribution first: 2(x + 3) = 16 becomes 2x + 6 = 16, then x = 5. Fractions are handled by clearing denominators: for x/4 + 3 = 8, subtract 3 (x/4 = 5), then multiply by 4 (x = 20).

Word problems follow a reliable routine: define the variable, write the equation, solve, and check the answer against the story. If a taxi charges a $3 pickup fee plus $2 per mile and the ride costs $17, let m be miles: 2m + 3 = 17, so 2m = 14 and m = 7 miles. The check is part of the solution — a negative number of miles or a fractional answer where only whole ones make sense means something went wrong.

Try it yourself: Solve: (a) 5x − 9 = 31, (b) x/3 + 4 = 10, (c) 3(x − 2) = 2x + 5. Answers: (a) x = 8; (b) x = 18; (c) 3x − 6 = 2x + 5, so x = 11.

Key vocabulary: Two-step equation — an equation requiring two inverse operations to isolate the variable; inverse operation — the operation that undoes another (addition/subtraction, multiplication/division); solution — the value that makes the equation true.

Lesson 9: Inequalities on the Number Line

An inequality compares values with < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike an equation, which usually has one solution, an inequality has a whole set of solutions — an interval on the number line. The statement x > 3 means every number greater than 3: 3.1, 4, 100, all of them. We graph it with an open circle at 3 (3 itself is not included) and shading to the right. For x ≥ 3 the circle is filled in, because 3 is included.

Solving inequalities works like solving equations, with one critical rule: when you multiply or divide both sides by a negative number, flip the inequality sign. For −2x < 10, dividing by −2 gives x > −5. Why? Multiplying by −1 reflects the number line across 0, reversing the order of every pair. Forgetting to flip is the single most common inequality error, so circle the moment a negative multiplier appears. Adding or subtracting never flips the sign.

Inequalities model real limits: speed limits (v ≤ 55), minimum ages (a ≥ 16 to drive), budgets (cost ≤ 200). Compound inequalities combine two conditions: 2 < x ≤ 6 means x is greater than 2 AND at most 6 — graphed as the segment between open circle at 2 and closed circle at 6. A statement like x < 1 or x > 5 graphs as two separate rays.

Try it yourself: Solve and describe the graph: (a) 2x + 5 < 13, (b) −4x ≥ 24, (c) 1 ≤ x + 3 < 7. Answers: (a) x < 4, open circle at 4, shade left; (b) x ≤ −6, closed circle at −6, shade left; (c) −2 ≤ x < 4.

Key vocabulary: Inequality — a comparison using <, >, ≤, or ≥; solution set — all values making the inequality true; compound inequality — two inequalities joined by AND or OR.

Lesson 10: The Coordinate Plane and Graphing Points

The coordinate plane is formed by two perpendicular number lines: the horizontal x-axis and the vertical y-axis, crossing at the origin (0, 0). Every point is named by an ordered pair (x, y): move x units horizontally from the origin (right if positive, left if negative), then y units vertically (up if positive, down if negative). The point (3, −2) is 3 right and 2 down. The axes divide the plane into four quadrants, numbered I through IV counterclockwise starting at the upper right.

Plotting points is the first step toward graphing relationships. A table of values becomes a picture: for the rule y = 2x + 1, the pairs (0, 1), (1, 3), (2, 5) lie on a straight line. If a point satisfies the rule, it sits on the graph; if not, it doesn't. The point (2, 4) fails y = 2x + 1 because 2(2) + 1 = 5, not 4. This test — substitute and check — is how you verify any point on any graph.

The coordinate plane also gives a distance shortcut: the distance between two points on the same horizontal or vertical line is just the difference of their coordinates. From (1, 5) to (6, 5) is 5 units; from (2, 1) to (2, 7) is 6 units. These ideas grow into the distance formula and the graphing of lines, curves, and data in every later course.

Try it yourself: Plot (5, −3) and (−4, −4); name their quadrants. Does (3, 7) satisfy y = 2x + 1? Answers: (5, −3) is Quadrant IV; (−4, −4) is Quadrant III; yes, 2(3) + 1 = 7.

Key vocabulary: Ordered pair — (x, y) locating a point; origin — (0, 0), where the axes cross; quadrant — one of the four regions of the coordinate plane.

Lesson 11: Area and Perimeter

Perimeter is the distance around a shape — a one-dimensional measurement in units like cm or inches. For a rectangle, P = 2(length + width); for a square, P = 4s. Area is the amount of surface enclosed, measured in square units (cm², in²). A rectangle's area is length × width. These two ideas answer different real questions: perimeter tells you how much fence to buy; area tells you how much grass seed to spread.

The triangle area formula, A = ½ × base × height, deserves a moment. Any triangle is exactly half of a parallelogram with the same base and height (two copies of the triangle form the parallelogram), and a parallelogram's area is base × height. The height must be the perpendicular distance — for a right triangle, the two legs serve as base and height, which makes right triangles easy. A parallelogram keeps A = b × h, and a trapezoid averages its two parallel sides: A = ½(b₁ + b₂) × h.

Circles need π (pi), the constant ratio of circumference to diameter, approximately 3.14159. Circumference C = πd = 2πr; area A = πr². Notice area uses r² while circumference uses r — mixing them up is the classic circle error. When a problem gives diameter, halve it first to get the radius. And always label units: a radius of 5 cm gives an area of 25π ≈ 78.5 cm².

Try it yourself: Find the area and perimeter of: (a) a triangle with base 10 cm and height 7 cm, (b) a circle with diameter 14 in (use π ≈ 3.14), (c) a trapezoid with bases 9 and 15 and height 8.

Lesson 12: Volume of Rectangular Prisms

Volume measures how much space a three-dimensional object fills, in cubic units (cm³, in³, ft³). For a rectangular prism (a box), volume is length × width × height: V = lwh. Why? The base layer holds l × w unit cubes, and there are h layers stacked on top. A box 5 cm long, 3 cm wide, and 4 cm tall holds 5 × 3 = 15 cubes per layer and 4 layers: 60 cm³. A cube is the special case where all sides are equal: V = s³.

Volume problems often run in reverse: given the volume and two dimensions, find the third. A fish tank with volume 240 liters (240,000 cm³), base 80 cm by 50 cm, has height 240,000 ÷ (80 × 50) = 60 cm. Unit conversion matters: 1 m³ = 1,000,000 cm³ because 1 m = 100 cm and 100³ = 1,000,000. Always convert all dimensions to the same unit before multiplying.

Volume connects to real decisions: how much water a pool holds, how many boxes fit in a moving truck, how much concrete a foundation needs. When boxes are packed into a larger box, count how many fit along each dimension separately (using division and rounding down), then multiply — leftover space in one direction cannot be borrowed by another.

Try it yourself: (a) Volume of a prism 7 m × 2 m × 3 m. (b) A box holds 360 cm³ with base 12 cm × 6 cm — its height? Answers: (a) 42 m³; (b) 360 ÷ 72 = 5 cm.

Key vocabulary: Volume — the amount of 3-D space an object occupies, in cubic units; rectangular prism — a box with rectangular faces; cube — a prism with all sides equal, V = s³.

Lesson 13: Surface Area and Nets

Surface area is the total area of all the faces of a three-dimensional solid. A net is a flat pattern that folds up into the solid, and it is the best tool for finding surface area: unfold the solid, find the area of each face, and add. For a rectangular prism with length l, width w, and height h, the surface area is 2lw + 2lh + 2wh, because opposite faces match in pairs. For a cube with side s, it is simply 6s².

For a cylinder, the net is two circles plus a rectangle. The circles have area πr² each, and the rectangle has height h and width equal to the circle's circumference, 2πr, so its area is 2πrh. The total surface area is therefore 2πr² + 2πrh. The lateral area is the area of the sides only, without the bases — for a prism, lateral area is the perimeter of the base times the height.

Real-world application: A painter estimating paint for a room, a manufacturer designing a cereal box, and a gift-wrapper cutting paper are all computing surface area. Wrapping paper must cover every face, and paint covers walls and ceiling — both are surface-area problems.

Try it yourself: Find the surface area of: (a) a cube with side 5 cm, (b) a rectangular prism 2 cm by 3 cm by 7 cm, (c) a cylinder with radius 3 cm and height 4 cm (leave in terms of π). Answers: (a) 6(25) = 150 cm²; (b) 2(6) + 2(14) + 2(21) = 82 cm²; (c) 2π(9) + 2π(3)(4) = 42π cm².

Key vocabulary: Net — a flat pattern that folds into a solid; surface area — total area of all faces; lateral area — area of the sides only, excluding bases.

Lesson 14: Angles, Triangles, and the Pythagorean Theorem

Angles are classified by measure: acute (less than 90°), right (exactly 90°), obtuse (between 90° and 180°), and straight (180°). The angles of any triangle add to 180°, so if two angles are known, the third is 180° minus their sum. Complementary angles add to 90°; supplementary angles add to 180°. These facts let you find unknown angles without measuring.

In a right triangle, the side opposite the right angle is the hypotenuse (the longest side), and the other two sides are the legs. The Pythagorean Theorem states that a² + b² = c², where a and b are the legs and c is the hypotenuse. It works in both directions: if a² + b² = c², the triangle is a right triangle. Whole-number triples satisfying the theorem, like (3, 4, 5) and (5, 12, 13), are called Pythagorean triples.

Real-world application: Builders use the 3-4-5 triangle to check that a corner is square: mark 3 feet along one wall and 4 feet along the other; the diagonal must be 5 feet. Ladders, ramps, and roof slopes all form right triangles whose lengths come from the Pythagorean Theorem.

Try it yourself: Solve: (a) A right triangle has legs 9 and 12. Find the hypotenuse. (b) A triangle has angles 35° and 90°. Find the third angle. (c) Is a triangle with sides 7, 24, 25 a right triangle? Answers: (a) 15; (b) 55°; (c) Yes, 49 + 576 = 625 = 25².

Key vocabulary: Hypotenuse — the longest side of a right triangle, opposite the right angle; legs — the two shorter sides; Pythagorean triple — whole numbers (a, b, c) with a² + b² = c²; complementary/supplementary — angle pairs summing to 90°/180°.

Lesson 15: Data Displays: Mean, Median, Mode, and Graphs

A data set can be summarized by its measures of center. The mean is the average: add all values and divide by how many there are. The median is the middle value when the data are in order (or the average of the two middle values for an even count). The mode is the most frequent value; a data set can have no mode or several. The range (max − min) measures spread. An outlier is a value far from the rest, and it pulls the mean more than the median.

Choosing the right display matters. A bar graph compares categories; a line graph shows change over time; a line plot shows individual values with x-marks; a histogram groups numerical data into intervals. A good display has a title, labeled axes, and a consistent scale. Misleading graphs usually break one of these rules — for example, a y-axis that does not start at zero can make small differences look huge.

Real-world application: Sports analysts compare players with batting averages (means) and medians; weather reports summarize a month of rainfall; schools report median test scores so one extreme score does not distort the picture. Every chart in the news is a data display to read critically.

Try it yourself: For 12, 15, 18, 15, 22, 30, find the mean, median, mode, and range. Which display best shows a city's temperature over 12 months? Answers: Mean = 112/6 ≈ 18.7; ordered 12, 15, 15, 18, 22, 30, median = (15 + 18)/2 = 16.5; mode = 15; range = 18. A line graph (change over time).

Key vocabulary: Mean — the average; median — the middle value; mode — the most frequent value; range — max minus min; outlier — a value far from the rest of the data.

Lesson 16: Probability Basics

Probability measures how likely an event is, as a number from 0 (impossible) to 1 (certain). For equally likely outcomes, P(event) = (favorable outcomes) / (total outcomes). Rolling a 4 on a fair die has probability 1/6; rolling an even number has probability 3/6 = 1/2. The sample space is the list of all possible outcomes. The probabilities of everything in the sample space add to 1, so P(not E) = 1 − P(E).

Theoretical probability comes from reasoning about equally likely outcomes; experimental probability comes from actual trials: (times the event happened) / (total trials). As trials increase, experimental probability usually gets closer to theoretical — the law of large numbers. For two events, P(A and B) for independent events is P(A) × P(B), and P(A or B) for mutually exclusive events is P(A) + P(B). To predict counts, multiply the probability by the number of trials: expected count = P × trials.

Real-world application: Weather forecasts are probabilities: a 30% chance of rain means rain occurred on about 30% of days with similar conditions. Game designers balance dice and card mechanics with probability, and insurance companies set premiums from the probability of claims.

Try it yourself: A spinner has 8 equal sections numbered 1-8. Find: (a) P(rolling an odd number), (b) P(number greater than 5), (c) P(not a 3). Answers: (a) 4/8 = 1/2; (b) 3/8; (c) 7/8.

Key vocabulary: Probability — a number 0 to 1 measuring likelihood; sample space — all possible outcomes; event — a set of outcomes; theoretical/experimental probability — from reasoning / from trials.

Lesson 17: Proportional Relationships and Unit Rates in Graphs

A relationship is proportional if the ratio of the two quantities is constant and the graph is a straight line through the origin. It can be written y = kx, where k is the constant of proportionality, equal to the unit rate (the value of y when x = 1). If a car travels 150 miles in 3 hours at a steady speed, k = 150/3 = 50 miles per hour, and the equation is d = 50t. The graph passes through (0, 0), (1, 50), (3, 150).

To test proportionality from a table, check that y/x is the same for every row (and that (0, 0) fits). To test it from a graph, check that the line is straight and passes through the origin. A line that is straight but misses the origin — like a phone bill with a fixed monthly fee plus a per-minute charge — is linear but NOT proportional, because there is a starting value other than zero.

Real-world application: Unit rates drive shopping (price per ounce), cooking (scaling recipes), travel (miles per gallon), and pay (dollars per hour). Comparing two phone plans or two bulk deals is a comparison of unit rates and proportionality.

Try it yourself: A cyclist rides 36 km in 2 hours at a steady pace. (a) What is the unit rate? (b) Write y = kx for distance. (c) How far in 5 hours? Answers: (a) 18 km/h; (b) d = 18t; (c) 90 km.

Key vocabulary: Proportional relationship — y = kx, a line through the origin; constant of proportionality — k, the unit rate; unit rate — the value of y when x = 1.

Lesson 18: Linear Relationships: Tables, Graphs, and Slope Preview

A linear relationship is one whose graph is a straight line. Its steepness is the slope: the change in y divided by the change in x, or “rise over run.” If y increases by 4 every time x increases by 2, the slope is 4/2 = 2. In the equation y = mx + b, m is the slope and b is the y-intercept, the value of y when x = 0. Every linear relationship can be described by a table, a graph, and an equation — and you should be able to move between all three.

To graph from a table, plot the points and check they line up; to build a table from an equation, pick x-values and compute y. The sign of the slope tells the story: positive slope means y grows as x grows; negative slope means y falls; zero slope means y never changes (a horizontal line). A slope of 0/undefined distinction matters: a vertical line has undefined slope because its run is zero and division by zero is impossible.

Real-world application: Phone plans, gym memberships, savings growth, and taxi fares are all linear relationships: a starting amount plus a steady rate. Reading the slope as a rate (dollars per week, miles per hour) turns a graph into a decision.

Try it yourself: A plant is 6 cm tall and grows 2 cm per week. (a) Write y = mx + b. (b) How tall after 7 weeks? (c) Is the relationship proportional? Why? Answers: (a) y = 2x + 6; (b) 20 cm; (c) No — the line does not pass through the origin because of the starting height 6.

Key vocabulary: Linear relationship — a relationship whose graph is a straight line; slope — rise over run, the rate of change; y-intercept — the value of y when x = 0.

Lesson 19: Equations with Variables on Both Sides — A First Look

Some equations have the variable on both sides, like 3x + 4 = x + 12. The strategy is the same as always: keep the equation balanced, and gather all variable terms on one side and all constants on the other. Subtract x from both sides: 2x + 4 = 12. Subtract 4: 2x = 8. Divide: x = 4. Always check by substituting back: 3(4) + 4 = 16 and 4 + 12 = 16. Balanced.

Three outcomes are possible. Most equations have one solution. If the variable disappears and a true statement remains (like 5 = 5), the equation is an identity with infinitely many solutions — every number works. If a false statement remains (like 2 = 7), there is no solution. Spotting these cases saves time: simplify first, and if the variable vanishes, read the remaining statement.

Real-world application: Comparing two streaming plans, two job offers, or two shipping options always produces an equation with the variable on both sides: the break-even point is where the two expressions are equal. Solving it tells you when to switch.

Try it yourself: Solve: (a) 7x + 2 = 3x + 14, (b) 3(x − 1) = 3x − 3, (c) 6x + 5 = 6x + 11. Answers: (a) x = 3; (b) identity, infinitely many solutions; (c) no solution.

Key vocabulary: Identity — an equation true for every value; no solution — an equation true for no value; break-even point — where two expressions are equal.

Lesson 20: Capstone: Real-World Projects with Pre-Algebra

Pre-algebra is a toolkit, and a capstone project is where the tools work together. Every project follows the same four steps: (1) understand the question and list what you know, (2) choose the right tools (equations, ratios, area, data), (3) compute carefully and keep units attached, (4) check the answer against reality — is it reasonable? A budget that adds to more than the income, or a paint estimate that covers a wall ten times over, signals a mistake somewhere.

Project 1 — Plan a trip. A family drives 420 miles. The car gets 28 miles per gallon and gas costs $3.50 per gallon. Gallons needed: 420 / 28 = 15; gas cost: 15 × 3.50 = $52.50. Add two nights at $95: total $52.50 + $190 = $242.50. Project 2 — Design a room. A bedroom 12 ft by 10 ft with 8-ft ceilings has floor area 120 ft² and wall area 2(12)(8) + 2(10)(8) = 352 ft². One gallon covers 350 ft², so one gallon (barely) covers the walls, or two for two coats.

Real-world application: Budgeting a trip, pricing a small business, planning a garden, and analyzing game statistics all combine fractions, percents, equations, geometry, and data in one task. The capstone habit — plan, compute, check — is the same habit used in every math course that follows.

Try it yourself: A rectangular garden is 9 ft by 6 ft. Fencing costs $4 per foot and soil costs $2 per square foot. What is the total cost? Answer: Perimeter = 30 ft, fencing = $120; area = 54 ft², soil = $108; total = $228.

Key vocabulary: Budget — a plan for income and spending; estimate — a reasonable approximation; scale — the ratio between a drawing and the real object.