MA001: College Algebra

1a. solve linear, rational, quadratic, radical, and absolute value equations in one variable

• Compare the general forms of linear equations, quadratic equations, and absolute value equations in one variable.
• Identify different methods for solving quadratic equations.
• Describe the difference between rational equations and radical equations in one variable.
• Identify strategies for solving rational and radical equations. 

Equations in a single variable can appear in a variety of forms, with different strategies used depending on the nature of the equation. Perhaps the simplest equation is a linear equation, which is an equation in which the variable is only to the first power. The typical form of a linear equation is $a x+b=0$, in which $a$ and $b$ are real numbers and $a \neq 0$. The solution to such an equation is $x=-\frac{b}{a}$.

In contrast, in a quadratic equation the variable is raised to the second power. The general form of a quadratic equation is $a x^{2}+b x+c=0$, where $a, b$, and $c$ are real numbers and $a \neq 0$. Quadratic equations can always be solved using the quadratic formula, $x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$.

Depending on the values of $a, b$, and $c$, it may be efficient to solve the quadratic equation by other methods. For instance, if the equation can be factored into $(x+h)(x+k)=0$, the solutions are found by solving $x+h=0$ and $x+k=0$. Completing the square involves rewriting the equation into the form $(x+h)^{2}=k$, so that the solution becomes $x=-h \pm \sqrt{k}$.

An absolute value equation is of the form $|a x+b|=c$. The solution to such an equation is $a x+b=c$ or $a x+b=-c$.

Rational equations contain at least one rational expression, which is a ratio of two polynomials. In contrast, radical equations contain a variable in the radicand. To solve a rational equation, multiply both sides of the equation by the least common denominator or rewrite the equation to be of the form $\frac{a}{b}=\frac{c}{d}$ and then solve $a d=b c$. To solve a radical equation, isolate the radical and then raise both sides of the equation to the index power of the radical; if the original equation contains more than one radical, this step might need to be repeated. For both rational and radical equations, check the potential solutions in the original equation to ensure they satisfy the original equation.

Review the material in Solving Linear Equations in One Variable, Solve Quadratic Equations by Factoring, and Equations That are Quadratic in Form.

1b. classify solutions to linear, rational, quadratic, radical, and absolute value equations

• When will a linear equation have no solutions? When will it have an infinite number of solutions?
  
• What is the discriminant and how can it be used to determine the nature of the roots of a quadratic equation?
  
• How is it possible to determine if an absolute value equation will have one, two, or no solutions? 
  
• What are extraneous solutions and when might they arise?

When solving equations, situations arise in which specific solutions exist, no solutions exist, or an infinite number of solutions exist. For instance, when solving a linear equation, if the result simplifies to $x=k$, then a specific solution exists. If the result simplifies to a true statement, such as $5=5$ or $x=x$, then there are an infinite number of solutions because every value of $x$ works in the equation. If the result simplifies to a false statement, such as $10=20$, then there are no solutions to the equation.

The discriminant for a quadratic equation is the value $b^{2}-4 a c$, which is the radicand in the quadratic formula. Its value can be used to determine the nature of the roots of the equation: two real roots if the discriminant is greater than 0 and those two roots are rational if the discriminant is a perfect square; one rational root if the discriminant is 0; and no real roots if the discriminant is negative.

The absolute value of a number is always non-negative, with $|x|=x$ when $x \geq 0$ and $|x|=-x$ when $x < 0$. So, the absolute value equation $|a x+b|=c$ has one solution if $c=0$, two solutions if $c > 0$, or no solutions if $c < 0$.

In both rational and radical equations, it is possible to obtain an extraneous solution, that is a solution to the resulting equation when the rational expressions or radicals are eliminated but that does not satisfy the original equation. This often occurs because the extraneous solution causes a denominator in the rational equation to equal zero or the radicand in a radical equation to be negative. So, when solving rational and radical equations, it is essential to check all potential solutions in the original equation.

1b. Review the material in Solving Linear Equations in One Variable, Solve Quadratic Equations by Factoring, and Equations That are Quadratic in Form.

1c. construct linear equations from words

• Identify phrases that are often used to indicate the four operations when translating a verbal sentence to a mathematical equation.

Many real-world situations are represented and modeled by equations. In a linear equation of the form $a x+b=c$, with $a, b$, and $c$ real numbers and $a \neq 0$, the coefficient a typically represents the change in each unit of the variable and $b$ represents a fixed amount in the situation. When translating from a verbal sentence to a mathematical sentence, it is important to identify the variable and its meaning to assist with interpreting the solution to the equation. For instance, consider the following context: An individual opens a savings account with an initial balance of $100 and plans to deposit $75 into the account each month. In how many months will the balance be $1600? The variable of interest is the number of months, which can be represented by the letter $m$. The fixed amount is $100 and represents $b$; the change for each unit of the variable (months) is $75 and represents $a$. So a reasonable equation to model this situation is $75 m+100=1600$.

Although many different words can be used to suggest the appropriate operation, there are some words that regularly appear:

Although many different words can be used to suggest the appropriate operation, there are some words that regularly appear: 

• Addition: sum, exceeds, is more than, increased by
• Subtraction: difference, decreased by, less, less than
• Multiplication: product, multiplied by, twice, three times
• Division: quotient, divided by

The verb is typically translates as the equal sign. 

It is important to watch for situations that indicate the need to group two or more terms in parentheses. Consider the wording and translation of the following two phrases:

• the sum of three times a number and four: $3 n+4$
• three times the sum of a number and four: $3(n+4)$

Review the material in Applications of Linear Equations.

1d. represent linear and absolute value inequalities using standard notation and graphs

• Compare representing an interval on a number line with an inequality, using set-builder notation, or using interval notation.
• Distinguish between intervals that include the endpoints and those that do not.
• What is a compound inequality?

Although a linear equation typically has a single solution, a linear inequality represents an interval on the number line. Consider the two inequalities below:

The first interval can be represented as $x \geq 2$. Notice that the endpoint of the ray is a closed circle to indicate that 2 is part of the solution. In set-builder notation, this inequality would be represented as $\{x: x \geq 2\}$. In interval notation, the inequality would be represented as $[2, \infty)$, with the bracket on the left used to indicate that 2 is part of the solution and the infinity sign, $\infty$, used to indicate that the solution contains all real numbers greater than or equal to 2.

In contrast, the second inequality can be represented as $x < -3$; notice that the open circle at the endpoint indicates that $-3$ is NOT part of the solution. The interval can also be represented as $\{x: x < -3\}$ or as $(-\infty,-3)$, using set-builder or interval notation, respectively. In interval notation, a parenthesis is used rather than a bracket to indicate the endpoint is NOT part of the solution.

A compound inequality contains more than one inequality within the statement, such as $0 < x \leq 10$ as well as $x < 0$ or $x > 10$. In an inequality such as $0 < x \leq 10$, one should always be able to hide the middle portion of the inequality and still have a true statement, namely $0 \leq 10$. This can ensure that one avoids writing statements that make no sense, such as $0 > x > 10$; if the middle inequality is hidden, the resulting inequality $0 > 10$ is meaningless.

Absolute value inequalities lead to compound inequalities. For instance, $|X| < B$ is equivalent to $-B < X < B$. In contrast, $|X| > B$ is equivalent to $X < -B$ or $X > B$.

Review the material in Absolute Value Inequalities.

1e. solve linear and absolute value inequalities

• Identify differences between solving linear inequalities and solving linear equations.

The techniques used in solving linear equations are also used in solving linear inequalities, with one major difference. When solving inequalities, the inequality sign reverses when multiplying or dividing by a negative. For instance, to solve $x+4 < 12$, one subtracts 4 from both sides of the inequality to obtain $x < 8$. In contrast, to solve $-4 x < 12$, one divides both sides of the inequality by $-4$ and then must reverse the inequality sign to obtain $x > -3$.

To solve an absolute value inequality, first remove the absolute value sign using the fact that $|X| < B$ is equivalent to $-B < X < B$ or that $|X| > B$ is equivalent to $X < -B$ or $X > B$. Then solve the compound inequalities using the same techniques as used when solving other inequalities.

Review the material in Absolute Value Inequalities.

1f. perform algebraic operations on complex numbers

• Define the imaginary number $i$ and explain how it is used to simplify square roots of negative numbers.
• Define a complex number and distinguish between the real part and the imaginary part.
• Explain how to perform the operations of addition, subtraction, and multiplication with complex numbers and represent the result in standard form.
• Define a complex conjugate of a complex number and explain how complex conjugates are used to express divisions of complex numbers in standard form.
• Explain how to simplify powers of $i$.

The imaginary number $\boldsymbol{i}$ is used to represent $\sqrt{-1}$ so that $i^{2}=-1$. When simplifying a square root with a negative radicand, the imaginary number is used to represent the negative, so that $\sqrt{-64}=8 i$ and $\sqrt{-12}=2 i \sqrt{3}$. A complex number in standard form is a number of the form $\mathbf{a}+\boldsymbol{b i}$, where $\boldsymbol{a}$ represents the real part and $\boldsymbol{b}$ represents the imaginary part.

Operations with complex numbers can be simplified using techniques similar to those used with other numbers. To add or subtract complex numbers, simply combine the real parts and then combine the imaginary parts. For instance, $(3+5 i)+(7-2 i)=(3+7)+(5-2) i=10+3 i$. To multiply complex numbers, apply the distributive property and then rewrite $i^{2}$ as $-1$. So, $(3+5 i)(7-2 i)=21-6 i+35 i-10 i^{2}=21+29 i-10(-1)=$ $31+29 i.$

Dividing two complex numbers and representing the result in standard form generally requires using the complex conjugate of $a+b i$, which is $a-b i$. That is, to find the complex conjugate of a complex number, find the additive inverse of the imaginary part. Then to find the quotient of two complex numbers, multiply numerator and denominator by the complex conjugate of the denominator and simplify as illustrated below.

$\frac{3+4 i}{2-7 i}=\frac{3+4 i}{2-7 i} \cdot \frac{2+7 i}{2+7 i}=\frac{6+21 i+8 i+28 i^{2}}{4+14 i-14 i-49 i^{2}}=\frac{-22+29 i}{4+49}=-\frac{22}{53}+\frac{29}{53} i$

The imaginary number $i$ has an interesting property related to its powers: $i^{1}=i ; i^{2}=-1 ; i^{3}=-i ; i^{4}=1$. The powers then repeat. So, to simplify a power of $i$, such as $i^{50}$, divide the exponent by $4$ and look at the remainder. So, $50 \div 4$ has a remainder of 2, meaning $i^{50}$ is equivalent to $i^{2}$ which is $-1$.

Review the material in Complex Numbers.

Unit 1 Vocabulary

This vocabulary list includes terms you will need to know to successfully complete the final exam.

• absolute value equation
• complex conjugate
• complex number
• compound inequality
• discriminant
• extraneous solution
• imaginary number, $i$
• imaginary part of a complex number
• linear equation
• quadratic equation
• quadratic formula
• radical equation
• rational equation
• real part of a complex number