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Description
Equation
78 linear first-degree equation in one variable $a\ne0$ \[ ax+b=c \] 99 quadratic second degree equation in standard form $a\ne0$ \[ ax^2+bx+c=0 \] 99 Solve Quadratic Equations, Zero Product Property $p,q\in\R$ If \[ pq=0 \] then \[ p=0\;\text{ or }\;q=0 \] 101 Solve Quadratic Equations, The Square Root Property $k\in\R$ If \[ x^2=k \] then \[ x=\pm\sqrt{k} \] 102 Solve Quadratic Equations, Complete the Square. Given $x^2+bx=c,$ add $\left(\frac{b}{2}\right)^2$ to both sides to form a perfect square trinomial on the left. Then, solve using the square root property. \[ \begin{align*} &x^2+bx=c\\ &\Rightarrow x^2+bx+\left(\frac{b}{2}\right)^2 =c+\left(\frac{b}{2}\right)^2\\ &\Rightarrow \left(x+\frac{b}{2}\right)^2 =c+\left(\frac{b}{2}\right)^2\\ &\Rightarrow x+\frac{b}{2} =\pm\sqrt{c+\left(\frac{b}{2}\right)^2}\\ &\Rightarrow x = -\frac{b}{2} \pm\sqrt{c+\left(\frac{b}{2}\right)^2}\\ \end{align*} \] 105 Solve Quadratic Equations, The Quadratic Formula. The equation must be in standard form. To derive the formula, complete the square. If \[ ax^2+bx+c=0\\ \] then \[ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \] 105 the discriminant. The term under the radical in the quadratic formula. \[ b^2-4ac \] 112 power equations \[ x^{m/n}=k \] 112 radical equation $u$ an algebraic expression and $k$ a real number. \[ \sqrt[n]{u}=k \] 118 quadratic type equation $a\ne0$ and $u$ an algebraic expression. \[ au^2+bu+c=0 \] 139 absolute value of $a$ is the distance on the real number line between zero and the number $a.$ $$ |a|= \left\{ \begin{array}{rl} a& &\text{ if } a\geq 0\\ -a& &\text { if } a\lt 0\\ \end{array} \right. $$ 140 absolute value equation If $x\geq0$ then \[ \abs{x} = x. \] If $x\lt0$ then \[ \abs{x} = -x. \] 141 properties of absolute value
  1. \[ \abs{ab}=\abs{a}\abs{b} \]
  2. \[ \left|{\frac{a}{b}}\right|=\frac{\abs{a}}{\abs{b}}(b\ne0) \]
  3. The triangle inequality \[ \abs{a+b}\leq\abs{a}+\abs{b} \]
  4. \[ \abs{a-b}\geq\abs{a}-\abs{b} \]
141 multiplication 141 division 141 triangle inequality \[ \abs{a+b}\leq\abs{a}+\abs{b} \] 142 absolute value equation \[ \abs{u}=\abs{v} \] if and only if \[ u=\pm v \] 144 absolute value inequality $u$ an algebraic expression, $k\in\R,$ $k\gt0.$ \[ \abs{u}\lt k \] if and only if \[ -k\lt u\lt k \] 144 absolute value inequality $u$ an algebraic expression, $k\in\R,$ $k\gt0.$ \[ \abs{u}\gt k \] if and only if \[ u\lt -k\;\text{or}\;u\gt k \]
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Notes
78 An equation is a statement declaring that two algebraic expressions are equal. An equation that becomes true when the variable is replaced by any permissible number is called an identity. An equation that is true only for some values of the variable, but not for others (or is never true for any value of the variable), is called a conditional equation. We refer to values of the variable that make the equation a true statement as the roots or solutions to the equation. We say that the root (solution) satisfies the equation. Equations that have the same roots are said to be equivalent equations. 78

Rules for generating equivalent equations

  1. Add the same expression to both sides of a given equation, or subtract the same expression from both sides.
  2. Multiply or divide both sides of a given equation by the same nonzero expression.
  3. Interchange the left-hand and right-hand sides of a given equation.
  4. Simplify algebraic expressions that appear on either side of a given equation.
78 To solve an equation means to find all values of the variable that make the equation a true statement. 79 To solve a linear equation apply the rules for generating equivalent equations to obtain an equivalent equation in which the variable is isolated on one side of the equation. The solution will appear on the other side. Check the solution by replacing the variable in the original equation with the derived solution to see if the equation becomes true. 80 Fractional equations reducible to linear form. An equation that contains one or more algebraic fractions is called a fractional equation. To eliminate the denominators, multiply both sides of the fractional equation by the least common denominator (LCD) for all the fractions in the equation. 81 If a fractional equation contains variables in the denominator, we need to check for division by zero. If we multiply both sides of a given equation by zero, the equation generated is no an equivalent equation. Thus, if we multiply both sides of a fractional equation by an LCD that contains a variable, we must check the solution to be certain that the LCD is indeed a nonzero quantity. If a found solution makes the LCD zero, it is called an extraneous root and should be discarded. 82 extraneous root. check for extraneous solutions when multiplying both sides of an equation by an expression containing a variable found in a denominator of the original equation. If the solution found makes any denominator zero, it is an extraneous root. 83 An equation that contains letters other than the variable for which we wish to solve is called a literal equation. 83 A formula is a mathematical or scientific rule in the form of a literal equation that describes a special relationship between two or more variables. Often, it is necessary to rearrange a formula and solve for one of its variables. 87

Solve Word Problems

  1. Read the problem carefully and determine what the question is asking you to find. Ask yourself what is known and what is unknown.
  2. Assign one of the unknowns the variable $x$ (or any other letter you wish), and express each of the other unknowns in terms of $x.$ When appropriate, draw a picture of the situation being described.
  3. Develop an equation that relates the known and unknown quanitites. This relationship may be given by an established formula or may require some intuition.
  4. Solve this equation for $x.$ Use this solution to answer the question the problem asked.
  5. Check your answer by making certain it satisfies the conditions of the problem.
88

Percent Mixture Problems

If a mixture of $b$ liters contains $r\%$ of a certain ingredient, then the amount $a$ of that ingredient (in liters) in the mixture is given. \[ a=\frac{r}{100}\cdot b \]

89-92

Examples of formulas and word problems

geometry problems

investment problems

simple interest formula

business problems

total cost

total cost = variable costs + fixed costs

total revenue

total revenue = price per unit $\times$ number of units sold

profit

profit = total revenue – total cost

93

Uniform Motion Problems

Formulas to find the distance traveled $d$ when given the constant rate of speed $r$ over a given time $t$.

\[ d=rt \qquad r=\frac{d}{t} \qquad t=\frac{d}{r} \] 94

Work Problems

If it takes $t$ hours to complete a certain job when working at a constant rate of speed, then $1/t$ of the job will be done in 1 hour.

105

Quadratic Equations - Nature of Roots and the Discriminant

Case 1

If $b^2-4ac\gt0,$ then $\sqrt{b^2-4ac}$ is a real number and the quadratic formula gives two distinct real roots: \[ \frac{-b+\sqrt{b^2-4ac}}{2a} \] and \[ \frac{-b-\sqrt{b^2-4ac}}{2a}. \] If $a,b,c$ are rational numbers and $b^2-4ac$ is the square of a rational number, then these two distinct real roots are rational numbers. Otherwise, the roots are irrational numbers.

Case 2

If $b^2-4ac\lt0,$ then $\sqrt{b^2-4ac}$ is a pure imaginary number and the quadratic formula gives two complex conjugate roots: \[ \frac{-b}{2a}+\frac{\sqrt{\abs{b^2-4ac}}}{2a}i \] and \[ \frac{-b}{2a}-\frac{\sqrt{\abs{b^2-4ac}}}{2a}i \]

Case 3

If $b^2-4ac=0,$ then $\sqrt{b^2-4ac}$ is zero and the quadratic formula gives only one real root: \[ -\frac{b}{2a}. \] The quantity $-\frac{b}{2a}$ is called a double root or repeated root of multiplicity two.

Summary of Discriminant Cases
Discriminant Nature of roots
$b^2-4ac\gt0$ Two distinct real roots
$b^2-4ac\lt0$ Two complex conjugate roots
$b^2-4ac=0$ One real root of multiplicity two

112

Power Equations - Nature of Real Roots

Case 1

If $k\gt0,$ $m>0,$ $m$ even, then $x^m=k$ has two real solutions, $\pm\sqrt[m]{k}.$

Example: $x^4=10\Rightarrow x=\pm\sqrt[4]{10}$

Case 2

If $k\lt0,$ $m>0,$ $m$ even, then $x^m=k$ has no real solution.

Example: $x^4=-10$ has no real solution.

Case 3

If $k\in\R,$ $m>0,$ $m$ odd, then $x^m=k$ has one real solution $\sqrt[m]{k}.$

Example: $x^5=18\Rightarrow x =\sqrt[5]{-18} =-\sqrt[5]{18}$

113

Solve Power Equations

To solve power equations of the form $x^{m/n}$ where $m$ and $n$ are positive integers that are relatively prime and $k\in\R,$ raise both sides to the $n\text{th}$ power to obtain $x^m=k^m.$ The equation $x^m=k^m$ can then be solved using previous methods.

However, does raising both sides of an equation to a positive integer power generate equivalent equations?

Consider \[ x=k\quad\text{ where }\quad k\in\R \] If we raise both sides to the $n\text{th}$ power, we obtain \[ x^n=k^n. \] There are two possibilities when solving this equation.

Case 1: $n\gt0,$ $n$ odd

\[ x=\sqrt[n]{k^n}\\ x=k \] equivalent equations are generated.

Case 2: $n\gt0,$ $n$ even

\[ x=\pm\sqrt[n]{k^n}\\ x=\pm k \] equivalent equations are not generated. since $-k$ does not satisfy the original equation. $-k$ is called an extraneous root.

Therefore, when raising both sides of an equation to a positive even integer power, check all solutions in the original equation to eliminate extraneous roots.

115

How to solve Radical Equations

Equations that contain the unknown in a radicand are called radical equations. To solve a radical equation containing one radical expression:

  1. Isolate the radical on one side of the equation.
  2. Eliminate the radical by raising both sides of the equation to the power equal to the index of the radical.
  3. If the index is even, check solutions for extraneous roots.
117

How to solve Factorable-type Equations

If one side of an equation can be factored into a product of algebraic expressions and the other side of the equation equals zero, then we may use the zero product property to help solve these factorable type equations.

118

Quadratic Type Equations - Ratio of Exponents $2:1$

Occurs when a quadratic-type equation, $au^2+bu+c=0,$ $a\ne0$ and $u$ an algebraic expression, has the characteristic that the exponent of $x$ in the first and second terms are in a $2:1$ ratio. For example, in the quadratic type equation \[ x^6-9x^3+8=0 \] the exponents $6$ and $3$ are in the ratio $2:1.$

118

Solve Quadratic-type Equations

Put it in the quadratic form $au^2+bu+c=0$ and solve it using the methods for solving a quadratic equation.

122

Inequalities

An inequality is a statement declaring one algebraic expression is less than $(\lt),$ $(\gt),$ $(\le),$ $(\ge),$ another algebraic expression. An inequality that becomes true no matter what real number is chosen to replace $x$ is called an absolute inequality. An inequality that is true only for some values of the variable but not for others (or is never true for any value of the variable) is called a conditional inequality. To solve an inequality means to find all values of the variables that make the inequality a true statement. We refer to these special values as the solution set of the inequality.

122

Intervals

Any portion of the real number line that corresponds geometrically to a line segment is called a bounded interval, and any portion of the real number line that corresponds geometrically to a ray (or the entire real number line) is called an unbounded interval. We use interval notation to describe the solution set of an inequality. The symbol $\infty$, read infinity, is not a real number. It is used to indicate that the solution interval has no right-hand boundary. Similarly, $-\infty$ is used to indicate an interval has no left-hand boundary.

Inequality Solution Interval Graph
Unbounded Intervals
$x\gt a$ $(a,\infty)$
$x\ge a$ $[a,\infty)$
$x\lt a$ $(-\infty, a)$
$x\le a$ $(-\infty, a]$
$-\infty\lt x \lt\infty$ $(-\infty,\infty)$
Open Interval
$a\lt x\lt b$ $(a,b)$
Closed Interval
$a\le x\le b$ $[a,b]$
Half-open Intervals
$a\lt x\le b$ $(a,b]$
$a\le x\lt b$ $[a,b)$
124 Inequalities that have the same solution set are said to be equivalent inequalities. 124

How to Generate Equivalent Inequalities

  1. Add or subtract the same quanitity on both sides of the given inequality.
  2. Multiply or divide both sides of a given inequality by the same positive quantity.
  3. Multiply or divide both sides of a given inequality by the same negative quantity and reverse the inequality (that is change $\lt$ to $\gt$, $\gt$ to $\lt$ $\le$ to $\ge$, $\ge$ to $\le$.)
  4. Interchange the left-hand and right-hand sides of a given inequality, and reverse the inequality.
  5. Simplify algebraic expressions that appear on either side of a given inequality.
126

Difference between equivalent inequalities and equations.

  1. Multiplying both sides of an inequality reverses the sense of the inequality.
  2. Interchanging left-hand and right-hand sides of an inequality reverses the sense of the inequality.
126 Inequalities of the form \[ ax+b\lt c \] \[ ax+b\le c \] \[ ax+b\gt c \] \[ ax+b\ge c \] where $a,b,c$ are real numbers ($a\ne0$), are referred to as linear inequalities. 126

How to solve a Linear Inequality

To solve a linear inequality, apply the rules for generating equivalent inequalities, and transform the given inequality into an equivalent one whose solution is obvious. 126

How to solve a Double Inequality

  1. To solve a double inequality in which the unknown appears only in the middle member, generate an equivalent double inequality by performing operations on all three members.
  2. To solve a double inequality in which the unknown appears in more than one member, write the double inequality $a\lt b\lt c$ as two inequalities \[ a\lt b\quad\text{ and }\quad b\lt c \] The solution set for such a double inequality is all real numbers common to the solution intervals of both $a\lt b$ and $b\lt c.$
130 polynomial inequality in standard form 130 critical values 131 critical value method 133 rational inequality in standard form 134 can't multiply both sides of an inequality with an expression containing the unknown. 134 solving rational inequalities 143 The solution set of the inequality $\abs{x}\lt k$, where $k$ is a real number, is the set of all real numbers that are less than $k$ units from zero. 143 The solution set of the inequality $\left|x\right|>k$, where $k$ is a real number, is the set of all real numbers that are greater than $k$ units from zero. 144 solving inequalities containing absolute values