Chapter 1: Equations and Inequalities Flashcards

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Questions and Answers

What is a linear equation?

highest power of x is 2

contains fractions

highest power of x is 1 (correct)

solves for multiple variables

What are the steps for solving a linear equation?

deal with fractions, 2. handle parentheses, 3. combine left/right, 4. move variables to one side and numbers to the other, 5. multiply or divide to isolate the variable.

What is the process for solving a word problem?

read two or three times, 2. assign variables to unknowns, 3. list facts, 4. translate to an equation, 5. solve, 6. check.

What is a quadratic equation?

highest power of x is 2 Signup and view all the answers

What methods can be used to solve a 2-term quadratic?

<ol> <li>factor out an x or 2. use the square root method.</li> </ol> Signup and view all the answers

What is the square root method?

For the equation x^2 = a, take the square root of both sides to find x = ±√a. Signup and view all the answers

What methods can be used to solve a 3-term quadratic?

<ol> <li>factor into two binomials or 2. use the quadratic formula.</li> </ol> Signup and view all the answers

What is the process for solving a 4-term quadratic?

<ol> <li>group the terms, 2. factor from each group, 3. factor the common factor, 4. set each factor to zero.</li> </ol> Signup and view all the answers

How do you handle a cubic equation (x^3)?

Factor out an x and use quadratic methods; the x gives one solution of 0. Signup and view all the answers

What is the process for solving a fourth degree equation (x^4)?

<ol> <li>let u = x^2, 2. make it 'quadratic', 3. solve the quadratic, 4. replace u with x^2, 5. solve.</li> </ol> Signup and view all the answers

What steps are involved in solving an equation with (x+a)^2?

<ol> <li>let u = (x+a), 2. make it 'quadratic', 3. solve the quadratic, 4. replace u with (x+a), 5. solve.</li> </ol> Signup and view all the answers

What is a radical equation?

<ol> <li>isolate the radical, 2. square both sides, 3. utilize linear or quadratic methods.</li> </ol> Signup and view all the answers

What are the two main concepts for complex numbers?

i^2 = -1; √(-1) = i. Signup and view all the answers

What is the conjugate of a + bi?

a - bi. Signup and view all the answers

How do you clear i out of a denominator?

Multiply the numerator and denominator by its conjugate. Signup and view all the answers

What is the process for equations with absolute values?

<ol> <li>isolate the absolute value, 2. remove and replace with ± on the opposite side, 3. solve two equations, changing only the RIGHT side.</li> </ol> Signup and view all the answers

What does < or > indicate?

open circles Signup and view all the answers

What does ≤ or ≥ indicate?

closed circles Signup and view all the answers

What is interval notation?

Uses ( ), [ ], ±∞, with no variables. Signup and view all the answers

What is set notation?

Uses < or >, ≤ or ≥, variables, not ±∞. Signup and view all the answers

What does the variable 'a' represent?

Variable 'a' is generally used as a placeholder in equations and expressions. Signup and view all the answers

Study Notes

Linear Equations and Solutions

A linear equation has the highest power of x equal to 1.
Solving a linear equation involves handling fractions, parentheses, combining like terms, isolating variables and constants, and finally getting the variable by itself through multiplication or division.

Word Problems

When solving a word problem, read carefully multiple times to understand the context.
Assign variables for unknowns, list known facts, translate the situation into an equation, solve it, and check the solution.

Quadratic Equations

A quadratic equation features the highest power of x as 2.
With a 2-term quadratic, utilize either the factoring out method or the square root method for solutions.
The square root method states that if ( x^2 = a ), then ( x = ±\sqrt{a} ).
A 3-term quadratic can be solved by factoring into two binomials or using the quadratic formula.
For 4-term quadratics, group the terms, factor from each group, factor the common elements, and set each factor to zero.

Higher Degree Equations

A cubic equation has a term in ( x^3 ) and can often be factored by removing an x to apply quadratic solving methods.
For a fourth-degree (quartic) equation, let ( u = x^2 ) to transform it into a quadratic. Solve and then replace ( u ) with ( x^2 ).

Special Equations

An equation of the form ( (x+a)^2 ) can similarly be handled by letting ( u = (x + a) ), turning it into a quadratic equation for resolution.
A radical equation requires isolating the radical, squaring both sides, and using linear or quadratic methods for further solving.

Complex Numbers

The two key concepts for complex numbers are that ( i^2 = -1 ) and ( \sqrt{-1} = i ).
The conjugate of a complex number ( a + bi ) is ( a - bi ).
To eliminate i from a denominator, multiply both the numerator and denominator by the conjugate.

Absolute Values

For equations containing absolute values, isolate the absolute value, replace with ± to form two separate equations to solve, noting that the opposite side remains unchanged.

Inequalities

Inequalities represented with < or > are graphed using open circles and parentheses.
Inequalities with ≤ or ≥ utilize closed circles and brackets.

Notation Types

Interval notation includes parentheses, brackets, and symbols like ±∞ without any variables.
Set notation incorporates inequalities and variables alongside ≤, ≥, <, or >.

Variable Representation

The letter "a" can denote a variable in various contexts within equations and expressions.

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Description

Test your knowledge of linear equations and solving techniques with these flashcards from Chapter 1. Each card focuses on essential definitions and problem-solving strategies for understanding equations and inequalities. Perfect for students looking to reinforce their learning.