Algebra I End-of-Course Comprehensive Study Guide

Representations of Inequalities and Solutions

Identifying the correct graphical representation of an inequality is a fundamental skill in Algebra I. For example, to solve the inequality $2x \ge -6$, one must divide both sides by $2$, yielding $x \ge -3$. On a number line, this solution is represented by a closed circle at $-3$ and a shaded region extending to the right, indicating all values greater than or equal to $-3$. In the coordinate plane, inequalities are often represented as shaded regions. For instance, the inequality $y < -2x + 1$ is represented by a dashed line (because the inequality is non-inclusive) with a slope of $-2$ and a y-intercept of $1$, with the area below the line shaded. Similarly, the expression $2x - y < 10$ can be rearranged into slope-intercept form as $y > 2x - 10$. This is graphed as a dashed boundary line with a slope of $2$ and a y-intercept of $-10$, with the shading occurring above the line.

Applications of inequalities are frequently used to express constraints in real-world scenarios. Diana's trip to Orlando covers a total of $182\,miles$. If she has already driven $x\,miles$, the remainder of the trip is $182 - x$. If her driving speed is constrained to below $70\,mph$, the time $t$ in hours required to complete the remainder of the drive is expressed by the inequality $t > \frac{182 - x}{70}$. In financial contexts, inequalities define budget limits. Mrs. Earle buying pillowcases for $2\,dollars$ ($x$) and sheets for $5\,dollars$ ($y$) where she buys twice as many pillowcases as sheets ($x = 2y$) and spends less than $40\,dollars$ results in the inequality $2(2y) + 5y < 40$, or $9y < 40$. This limits the maximum number of items she can purchase.

Functions and Functional Relationships

A functional relationship is defined by the rule that every input $x$ must correspond to exactly one output $y$. In tabular representations, a table does not represent a function if a single $x$-value is paired with multiple distinct $y$-values. For example, a table containing the pairs $(-1, -9)$ and $(-1, 9)$ violates this rule because the input $-1$ has two different outputs. Functions can be expressed in standard notation, where $y = -x + 3$ is equivalent to $f(x) = -x + 3$. This notation allows for the evaluation of specific values, such as using the function $f(x) = \frac{3 - x^2}{3 - x}$ to find $f(2) = \frac{3 - (2)^2}{3 - 2} = \frac{3 - 4}{1} = -1$.

In practical applications, functions model physical phenomena and business costs. The average daily high temperature for May in Ocala, Florida, is modeled by $f(n) = 0.2n + 80$, where $n$ is the day of the month. To find the maximum temperature on May 31st, one calculates $f(31) = 0.2(31) + 80$, which equals $86.2\,degrees$. In business, the cost of joining a Garden Club is represented by a function with an initial membership fee plus a per-item cost. If the Garden Club charges a $25\,dollar$ fee and $10\,dollars$ per rosebush, while a local nursery charges $15\,dollars$ per rosebush without a fee, the break-even point occurs where $10x + 25 = 15x$. Solving this shows that Dr. Chait starts saving money after purchasing more than $5$ rosebushes.

Manipulating Literal Equations and Formulas

Literal equations are formulas where variables represent specific physical or mathematical properties. To rewrite the perimeter of an isosceles triangle ($24 = 2s + b$) in terms of the congruent side length $s$, one isolates the variable to find $s = \frac{24 - b}{2}$. Another example is the pressure formula $P = \frac{1.2W}{H^2}$. Solving for the heel width $H$ involves multiplying by $H^2$, dividing by $P$, and taking the square root, resulting in $H = \pm \sqrt{\frac{1.2W}{P}}$. In physics, the distance formula $s = ut + \frac{1}{2}at^2$ can be solved for acceleration $a$ by first subtracting $ut$, multiplying by $2$, and dividing by $t^2$, yielding $a = \frac{2s - 2ut}{t^2}$.

Body Mass Index (BMI) calculations use the formula $B = \frac{703w}{h^{2}}$, where $w$ is weight and $h$ is height. To isolate weight, the equation is rearranged to $w = \frac{Bh^2}{703}$. For geometric shapes, the surface area $S$ of a cylinder is expressed as $S = 2\pi rl + 2\pi r^2$. Solving for the length $l$ requires subtracting the $(2\pi r^2)$ term and dividing by $2\pi r$, leading to $l = \frac{S - 2\pi r^2}{2\pi r}$. Similarly, the area of a parallelogram is given as $35pq\,square\,units$. If the base is $5pq^2\,units$, the height is found by dividing the area by the base: $\frac{35pq}{5pq^2} = \frac{7}{q}$, or $7p^0q^{-1}$.

Linear Equations, Slope, and Rates of Change

Linear equations describe relationships with a constant rate of change, known as the slope ($m$). The slope can be calculated from two points $(x_1, y_1)$ and $(x_2, y_2)$ using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$. For instance, Ryan's writing rate is determined from a graph showing $60\,sentences$ written in $40\,minutes$, which simplifies to a rate of $1.5\,sentences\,per\,minute$. In coordinate geometry, the slope of the line $2x - 5y = 10$ is found by converting it to $y = \frac{2}{5}x - 2$, identifying the slope as $\frac{2}{5}$. Parallel lines share the same slope, while perpendicular lines have slopes that are negative reciprocals of each other. The line perpendicular to $3y = 2x - 4$ (which has a slope of $\frac{2}{3}$) must have a slope of $-\frac{3}{2}$. Combined with a y-intercept of $3$, the equation becomes $y = -1.5x + 3$, or $2y = -3x + 6$.

Linear models often include initial values (y-intercepts) and constant additives. Alyssa gives speeches that increase in length by $30\,seconds$ each week ($150, 180, 210, 240\,s$). The length $L$ after $n$ weeks follows the sequence $L = 150 + 30(n - 1)$. To find the week she gives a $12-minute$ ($720\,second$) speech, the equation $720 = 150 + 30(n - 1)$ is solved for $n$, resulting in $n = 20$. Other linear models include phone bills ($t = 25 + 0.07m$), population growth ($15,400 + 325n = 18,000$), and depreciation, such as a computer valued at $1,200\,dollars$ losing $140\,dollars$ per year ($V = 1,200 - 140t$).

Systems of Equations and Direct Variation

A system of equations consists of two or more equations with the same variables. They can be solved using substitution or elimination. For the system $3x - 2y = 12$ and $4x - y = 11$, one can solve the second equation for $y = 4x - 11$ and substitute it into the first to find $x = 2$ and $y = -3$. Systems are also used to solve multi-variable word problems, such as determining the cost of adult ($x$) and student ($y$) tickets. If $5x + 2y = 48$ and $3x + 2y = 32$, subtracting the equations eliminates $y$, showing $2x = 16$, thus adult tickets cost $8\,dollars$.

Direct variation occurs when one variable is a constant multiple of another, expressed as $y = kx$. A farmer harvests potatoes in direct proportion to plants: $189\,potatoes$ from $9\,plants$ gives a constant $k = \frac{189}{9} = 21$. If he plants $14\,plants$, he expects to harvest $21 \times 14 = 294\,potatoes$. Inverse relationships also exist, such as the time to fill a fish tank. If it takes $40\,minutes$ at $3\,gallons\,per\,minute$, the volume is $120\,gallons$. Filling at $4\,gallons\,per\,minute$ would take $120 / 4 = 30\,minutes$.

Set Theory and Venn Diagrams

Set theory involves the study of collections of objects. The cross product of two sets $A$ and $B$, denoted $A \times B$, is the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$. If $A = \{0, 1, 2, 3, 4, 5\}$ and $B = \{6, 7, 8, 9\}$, there are $6 \times 4 = 24$ distinct pairs in the cross product. Intersection (denoted by $\cap$) finds elements common to both sets. For $A = \{f, c, a, t\}$ and $B = \{f, l, a\}$, the intersection $A \cap B$ is $\{f, a\}$.

Venn diagrams visually represent relationships between sets. In a group of student activities, if there are $32$ students total, with $15$ in basketball and $23$ in drama, and all students participate in at least one, the number of students in both is found by $(15 + 23) - 32 = 6$. In a more complex example involving keyword searches on Google, if $43\,million$ websites contain both "math" and "education" and $858\,million$ contain only "math", the total number of websites containing the word "math" is $901\,million$. Venn diagrams are also used to track participation in activities, such as girls at a camp choosing between volleyball and swimming, where subtracting the intersection and those who did neither identifies specific group sizes.

Exponents, Radicals, and Polynomial Simplification

Simplifying algebraic expressions requires the use of exponent rules and the distributive property. The rule for multiplying powers with the same base is $x^a \cdot x^b = x^{a+b}$. Consequently, $x^{5}x^{2} = x^{7}$, which is the same as $x^{4}x^{3}$. For division, $\frac{x^{a}}{x^{b}} = x^{a-b}$. This is applied in simplifying $\frac{(2x^{2})(8x^{6})}{4x^{6}} = \frac{16x^{8}}{4x^{6}} = 4x^{2}$. When binomials are added or subtracted, like terms are combined. For example, $(4x^2 - 2x + 8) - (x^2 + 3x - 2)$ simplifies to $3x^2 - 5x + 10$.

Radical expressions can be simplified by combining like radicals or factoring out perfect squares. The expression $4\sqrt{2} + 3\sqrt{2} - 5\sqrt{2}$ simplifies directly to $2\sqrt{2}$. Multi-step equation solving often involves the distributive property; for example, solving $3(2x + 6) - 4 = 14$ requires distributing the $3$ to both terms inside the parentheses to get $6x + 18 - 4 = 14$ before further simplification. Errors often occur in these initial steps, such as forgetting to distribute to the second term or incorrectly applying properties of equality.

Quadratics and Factoring Techniques

Quadratic equations are second-degree polynomials typically written as $ax^{2} + bx + c = 0$. Factoring is a primary method for solving these equations. The difference of squares is a specific factoring pattern: $a^{2} - b^{2} = (a - b)(a + b)$. Applying this to $y^{4} - 36$ results in $(y^{2} - 6)(y^{2} + 6)$. For a polynomial like $36y - 81x^{2}y$, one first factors out the greatest common factor $9y$, leaving $9y(4 - 9x^{2})$, which further factors into $9y(2 - 3x)(2 + 3x)$.

Solving quadratic equations like $x^{2} - 7x + 10 = 28$ involves setting the equation to zero: $x^{2} - 7x - 18 = 0$. This factors into $(x - 9)(x + 2) = 0$, giving solutions $x = 9$ and $x = -2$. If an equation is not easily factorable, the quadratic formula is used: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. For the equation $x^{2} + 8x + 7 = -8$, it must be rewritten as $x^{2} + 8x + 15 = 0$ before substituting $a = 1, b = 8, c = 15$ into the formula. This results in $x = \frac{-8 \pm \sqrt{8^2 - 4(1)(15)}}{2(1)}$. The path of physical objects, like a bird flying, can be modeled by these quadratic functions, and finding the zeros of the function ($f(x) = x^{2} - 4x - 32$) provides key information about its trajectory.