Equations and Inequalities Flashcards
Linear Equation
Generating Equivalent Equations
Solving a Linear Equation
- Simplify the algebraic expression on each side by removing grouping symbols and combining like terms.
- Collect all the variable terms on one side and all the numbers, or constant terms, on the other side.
- Isolate the variable and solve.
- Check the proposed solution in the orignal equation.
Rectangular Coordinate System (Cartesian Coordinate System)
Two number lines that intersect at right angles and their zero points.
X-Axis
The horizontal number line of the rectangular coordinate system.
Y-Axis
The vertical number line of the rectangular coordinate system
Origin
The point of intersection of the x-axis and the y-axis at their zero points
Quadrants
The division of the planes of the rectangular coordinate system into 4 quadrants.
- Upper Right
- Upper Left
- Lower Left
- Lower Right
Ordered Pair
A pair of real numbers that correspont to a point in the rectangular coordinate system. Written as ( x , y ).
x-coordinate
The first number of an ordered pair that denotes the distance and direction from the origin along the x-axis.
y-coordinate
The second number of an ordered pair that denotes the vertical distance and direction from the origin along the y-axis.
Plot
The points in the rectangular coordinate system that correspond to the location of an ordered pair.
x-intercept
The x-coordinate of a point where the graph intersects the x-axis
y-intercept
The y-coordinate of a point where the graph intersects the y-axis
Line Graphs
Used to illustrate trends over time
Linear Equation
Solving an Equation
The process of determining the values of the variables in an equation that result in a true statement when substituted into an equation.
Solution Set
The set of all solutions that satisfy an equation.
Equivalent Equations
Two or more equations that have the same solution set
Generating Equivelant Equations
Solving a Linear Equation
- Simplify the algebraic expression on each side by removing grouping symbols and combining like terms.
- Collect all the variable terms on one side and all the numbers or constant terms on the other side.
- Isolate the variable and solve.
- Check the proposed solution in the original equation.
Rational Equation
An equation containing one or more rational expressions.
Example - 1/x = 1/5 + 3/2x
Identity
An equation that is true for all real numbers for which both sides are defined.
Example - x + 3 = x + 2 + 1
Conditional Equation
An equation that is not an identity, but that is true for at least one real number
Inconsistent Equation
An equation that is not true for even one real number
Strategy for Solving Word Problems
- Read the problem careully several times until you can state in your own words what is given and what the problem is looking for. Let x (or any viariable) representone of the unknown quantities in the problem.
- If necessary, write the expressions for any other unknown quantities in the problem in terms of x.
- Write an equation in x that models the verbal conditions of the problem.
- Solve the equation and answer the problem’s question.
- Check the solution in the original wording of the problem, not in the equation obtained fromthe words.
Simple Interest Formula
I = Pr
- I = Simple Interest
- P = Principal amount
- r = Interest Rate (decimal)
Imaginary Unit i
Complex Numbers and Imaginary Numbers
The set of all numbers in the form
a + bi,
with real numbers a and b, and i, the imaginary unit, is called the set of complex numbers. The real number a is called the real part and the real number b is called the imaginary part of the complex number a + bi. If b <> 0, then the complex number is called an imaginary number. An imaginary number in the form bi is calle a pure imaginary number.
Equality of Complex Numbers
a + bi = c + di if and only if a = c and b = d
Adding and Subtracting Complex Numbers
- ( a + bi ) + ( c + di ) = ( a + c ) + ( b + d )i
In words, this says that you add complex number by adding their real parts, adding their imaginary parts, and expressing the sum as a complex number. - ( a + bi ) - ( c + di ) = ( a - c ) + ( b - d )i
In words, this says that you subtract complex numbers by subtracting their real parts, subtracting their imaginary parts, and expressing the difference as a complex number.
Conjugate of a Complex Number
The complex conjugate of the number a + bi is a - bi, and the complex conjugate of a - bi is a + bi. The multiplication of complex conjugates gives a real number.
( a + bi ) ( a - bi ) = a2 + b2
( a - bi ) ( a + bi ) = a2 + b2
Principal Square Root of a Negative Number
Quadratic Equation in General Form
A quadratic equation in x is an equation that can be written in the general form
ax2 + bx + c = 0
where a, b, and c are real numbers, with a <> 0. A quareatic equation in x is also called a second-degree polynomial equation in x.
Zero-Product Principle
If the product of two algebraic expressions is zero, then at least one of the factors is equal to zero.
If AB = 0, then A = 0 or B = 0
Solving a Quadratic Equation by Factoring
- If necessary, rewrite the equation in the general form of ax2 + bx + c = 0, moving all nonzero terms to one side, thereby obtaining zero on the other side.
- Factor completely.
- Apply the zero-product principle, setting each factor containing a variable equal to zero.
- Solve the equations in step 3.
- Check the solutions in the original equation.
The Square Root Property
Completing the Square
Solving a Quadratic Equation by Completing the Square
To solve ax2 + bx + c = 0 by completing the square:
- If a, the leading coefficient, is not 1, divide both sides by a. This makes the coefficient of the term x2-term 1.
- Isolate the variable terms on one side of the equation and the constant term on the other side of the equation.
- Complete the square.
- Add the square of half the coefficient of x to both sides of the equation.
- Factor the resulting perfect square trinomial.
- Use the square root property and solve for x.
The Quadratic Formula
Discriminant
The quantity b2 - 4ac, which appears under the ratical sign in the quadratic formula.
The Discriminant and the Kinds of Solutions to ax2 + bx + c = 0
Determining the Most Efficient Technique to Use when Solving a Quadratic Equation
The Pythagorean Theorem
Solving Radical Equations Containing nth Roots
- If necessary, arrange terms so that one radical is isolated on one side of the equation.
- Raise both sides of the equaiton to the nth power to eliminate the isolated nth root.
- Solve the resulting equation. If this equation still contains radicals, repeat steps 1 and 2.
- Check all proposed solutions int he original equation.
Solving Radical Equations in the Form of xm/n = k
Substitutions to get Equations into Quadratic Form
Quadratic in Form
An equation that can be expressed as a quadratic equation using an appropriate substitution
Rewriting an Absolute Value Equation without Absolute Value Bars
If c is a positive real number and u represents any algebraic expression, then | u | = c is equivalent to u = c or u = -c.
Open Interval (Notation)
Closed Interval (Notation)
Infinite Interval (Notation)
Intervals on the Real Number Line
Finding Intersections and Unions of Two Intervals
- Graph each interval on a number line.
- -
- 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 number in the two graphs.
Properties of Inequalities
Solving an Absolute Value Inequality
If u is an algebraic expression and c is a positive number
- The solutions of | u | < c are the numbers that satisfy -c < u < c.
- The solutions of | u | > c are teh numbers that satisfy u < -c or u > c
These rules are valid if < is replaces by <= and > is replaced by >=.
