Unit 1 Overview

When you have finished this unit, you will be able to do the following:

Solve linear equations in one variable

Solve linear equations containing fractions

Solve rational equations with variables in the denominators

Recognize identities, conditional equations, and inconsistent equations

Solve applied problems using mathematical models

Add and subtract complex numbers

Multiply complex numbers

Divide complex numbers

Perform operations with square roots of negative numbers

Solve quadratic equations by

• factoring
• square root property
• completing the square
• quadratic formula

Use the discriminant to determine the number and type of solutions

Determine the most efficient method to use when solving a quadratic equation

Solve problems modeled by quadratic equations

Solve polynomial equations by factoring

Solve radical equations

Solve equations with rational exponents

Solve equations that are quadratic in form

Solve equations involving absolute value

Solve problems modeled by equations

Use interval notation

Find intersections and unions of intervals

Solve linear inequalities

Recognize inequalities with no solution or all real numbers as solutions

Solve compound inequalities

Solve absolute value inequalities

The procedure for solving a linear equation is as follows:

1. Simplify the algebraic expression on each side by removing grouping symbols and combing like terms.
2. Collect all the variable terms on one side and all the numbers or constant terms on the other side.
3. Isolate the variable and solve.
4. Check the proposed solution in the original equation.

If an equation contains fractions, begin by multiplying both sides by the least common denominator, thereby clearing all the fractions.

The imaginary unit, i, is defined as

The set of numbers in the form a + bi is called the set of complex numbers where a is the real parts and b is the imaginary part. If b = 0, then the complex number is a real number. If b does not equal 0, then the complex number is an imaginary number. Complex numbers in the form of bi are called pure imaginary numbers.

Adding complex numbers

(a + bi) + (c + di) = (a + c) + (b + d)i

Subtracting complex numbers

(a + bi) - (c + di) = (a - c) + (b - d)i

Pattern involving imaginary numbers:

To multiply complex numbers, multiply as if they are polynomials. After completing the multiplication, replaced the i squared with -1 and simplify.

The complex conjugate is a + bi is a - bi and vice versa. The multiplication of complex conjugates gives a real number:

To divide complex numbers, multiply the numerator and the denominator by the complex conjugate of the denominator.

When performing operations with square roots of negative numbers, begin by expressing all square roots in terms of i. The principal square root of - b is defined by:

A quadratic equation in x can be written in the general form of:

To solve a quadratic equation, follow the following steps:

1. If necessary, rewrite the equation in the general form.

2. Factor completely.

3. Solve each set of parentheses for x by setting them equal to 0.

4. Check the solution(s).

The square root property will generate 2 answers and is as follows:

All quadratic equations can be solved by completing the square.

The quadratic formula will always give all solutions to a quadratic equation, whether the solutions are real or imaginary.

The discriminant indicates the number and type of solutions to the quadratic equation.

Some polynomial equations of degree 3 or greater can be solved by moving all terms to one side, obtaining zero on the other side, factoring, and using the zero-product principal. Factoring by grouping is often used as well.

A radical equation is an equation in which the variable occurs in a square root, a cube root, and so on. A radical equation can be solved by isolating the radical and raising both sides of the equation to a power equal to the radical's index. When raising both sides to an even power, check all proposed solutions in the original equation. Eliminate extraneous solutions from the solution set.

A radical equation with a rational exponent can be solved by isolating the expression with the rational exponent and raising both sides of the equations to a power that is the reciprocal of the rational exponent.

Steps of Solving a Radical Equation

• Assume that m and n are positive integers and m/n is in lowest terms.
• Isolate the expression with the rational exponent.
• If m is even, then you will have 2 answers.
• If m is odd, you will only have 1 answer.
• Check all proposed solutions in the original equation to find out if they are actual solutions or extraneous solutions.

An equation is in quadratic for if it can be written in the form of:

Absolute value equations in the form /u/ = c, can be solved by rewriting the equation without absolute value bars. u = c and u = -c

Solution sets of inequalities are expressed using set-builder notation and interval notation. In interval notation, parentheses indicate endpoints that are not included in an interval. Square brackets indicate endpoints that are included in an interval.

A procedure for finding intersections and unions of intervals is as follows

1. Graph each interval on a number line.

2. Interpret the results.

• To find the intersection, take the portion of the number line that the two graphs have in common.
• To find the union, take the portion of the number line representing the total collection of numbers in the two graphs.

A linear inequality can be expressed in four different ways

The procedure to solving a linear inequality is just like solving a linear equation, except if you multiply or divide by a negative number, you must flip the inequality sign.

Inequalities involving absolute value can be solved by rewriting the inequalities without absolute value bars. The c values, however, must be a positive number.

/u/ < c, then the inequality will look like -c < u < c. This will be a bounded answer.

/u/ > c, then the inequality will like like u < c or u > c