Solving Equations. Is a statement that two algebraic expressions are equal. EXAMPLES 3x – 5 = 7, x 2 – x – 6 = 0, and 4x = 4 To solve a equation in x.

Review – Section 1Solving Equations

Equation

Is a statement that two algebraic expressions are equal.

EXAMPLES

3x – 5 = 7, x2 – x – 6 = 0, and 4x = 4

To solve a equation in x means to find all real values of x for which the equation is true.

Domain

The set of real numbers for which an algebraic expression is defined.

Remember the real numbers are made up of the Natural, Whole, Integer, Rational and Irrational numbers. The real numbers can be represented on a number line with 0 as the origin.

Identity

An equation that is true for every real number in the domain of the variable

EXAMPLE

x2 – 9 =(x-3)(x+3) is an Identity, because it is a true statement for any real value of x

Conditional Equations

An equation that is true for some (or even none) of the real numbers in the domain of the variable.

EXAMPLE

x2 – 9 =0 is conditional because x =3 and x = -3 are the only values in the domain that satisfy the equation.

Linear Equation

An equation in one variable that can be written in the standard form:

ax + b = 0 where and a and b are real numbers and a ≠ 0

EXAMPLE

2x – 5 = 0

Solving Linear Equations

A linear equation has exactly one solution and the solution is found by isolating the variable on one side of the equation by a sequence of equivalent equations.

EXAMPLE2x – 4 = 102x – 4+ 4 = 10+ 42x = 14½(2x) = ½(14)X = 7

Extraneous Solution

A solution that does not satisfy the original equation.

EXAMPLE1/(x-2)= 3/(x+2) – 6x/(x2 -4)x = -2However, x = -2 yields a denominator of

zero, so the original equation has no solution.

Quadratic Equations

An equation in one variable written in the general form:

ax2 + bx + c = 0Where a, b, and c are real numbers and a ≠ 0

EXAMPLEx2 –x – 6 = 0

Solving Quadratic Equations

A quadratic equation has either one real solution, two real solutions or two conjugate imaginary solutions.

METHODS Factoring Square Root Principle Completing the Square Quadratic Formula

Factoring

EXAMPLEX2 –x – 6 = 0(x -3)(x+2) = 0X – 3 = 0 or x + 2 = 0X = 3 x = -2

Square Root Principle

EXAMPLE(x +3)2= 16x + 3 = ± 4X+3 = 4 or x + 3 = - 4X = 1 x = -7

Completing the Square

EXAMPLEx2 + 6x = 5x + 6x + (6/2)2 = 5 + (6/2)2 (x+3) 2 = 5 + 32

x + 3 = ±14X = -3 ±14

Quadratic Formula

EXAMPLE

2x2 + 3x -1 = 0x = {(-3 ±[( 32 -4(2)(-1))]½}/(2(2))x = ¾ ± 17/4

Other Types of Equations Polynomials of Higher Degree

3x4 = 48x2

x3 – 3x2 – 3x + 9 = 0 Equations Involving Radicals

(2x +7) ½ - x = 2(x – 4)⅔ = 25

Equations with Absolute Values|x – 2| = 3|x2 – 3x| = -4x + 6

Review – Section 2Solving Inequalities

Solving Linear Inequalities

Solve as if it were an equality, but remember to reverse the sign of the inequality whenever you multiply or divide by a negative number. Write the answer using interval notation.

EXAMPLE5x – 7 > 3x + 9x > 8 so the solution set is (8, ∞) meaning 8 is

not part of the solution, but all real numbers larger than 8 to infinity is the solution

Solving a Double Linear InequalitySolve as if it were an equality, but isolate

the variable as the middle term or solve as two separate inequalities

EXAMPLE-3 ≤ 6x – 1 < 3-⅓ ≤ x <⅔Solution interval is [-⅓, ⅔)[ ] means closed interval; ( ) means open

interval

Inequalities Involving Absolute Values

Write equivalent inequalities and solve.EXAMPLE

|x + 3| ≥ 7 (a) x + 3 ≥ 7 or (b) – (x + 3) ≥7 x ≥ 4 or x ≤ - 10 The solution interval is (-∞, -10] [4, ∞)

Solving a Polynomial Inequality

Write equivalent inequalities and solve.EXAMPLE

x2 – x – 6 < 0 Solve as if equality and find the critical values(x -3) (x + 2) = 0 then x =3, or x = -2 are the critical

values Check intervals(- ∞, -2) if x = -3 then (-3)2 –(-3) – 6 is positive(-2, 3) if x = 0 then (0) 2 – (0) – 6 is negative(3, ∞) if x = 4 then (4) 2 - (4) – 6 is positive

The solution interval is (-2, 3)

Solving a Rational Inequality

Find the critical values which occur for values making the numerator 0 or the denominator undefined. (right hand side must be 0 first)

EXAMPLE(2x – 7) / (x – 5) ≤ 3 Simplify to (-x +8)/(x-5) ≤ 0, thus critical values are

8 and 5Check the intervals (-∞, 5) if x = 4 then inequality is negative(5,8) if x = 6 then inequality is positive(8, ∞) if x = 9 then inequality is negativeThe solution interval is (-∞, 5) [8, ∞)

Finding the Domain of an ExpressionRemember the domain is the set of all x

values for which the expression is defined

EXAMPLE

(64 -4x2)½ Since this is a square root expression, the expression must be larger than 0. So find the critical values for (64 -4x2)½ ≥ 0 and test each of the intervals to find the solution set. The critical numbers are 4 and - 4

Review – Section 3Graphical Representation of Data

Coordinate Plane

consists of two perpendicular number lines, dividing the plane into four regions called quadrants

Parts of the Coordinate Plane

X-AXIS - the horizontal number line

Y-AXIS - the vertical number line

ORIGIN - the point where the x-axis and y-axis cross

Ordered Pair

ORDERED PAIR - a unique assignment of real numbers to a point in the coordinate plane consisting of one x-coordinate and one y-coordinate

(-3, 5), (2,4), (6,0), (0,-3)

Coordinate Plane

Distance Formula

D = [(x2 – x1)2 + (y2 – y1)2]½

EXAMPLE

(-2,1) and (3,4)

D = 34

Verfiy a Right Triangle

Use the distance formula and Pythagorean Theorem to determine a right triangle.

EXAMPLE

(2,1), (4,0) and (5,7)

Midpoint Formula

Find the midpoint of a line segment that joins two points in a coordinate plane.

M = [(x1 + x2)/ 2 ,(y1 + y2)/2]

EXAMPLE(-5,-3) and (9,3) M = (2,0)

Review – Section 4Graphs of Equations

Intercepts of a Graph

Solutions points that have zeros as either the x-coordinate or the y-coordinate are called intercepts because they are the points where the graph intersects the x-axis or y-axis

Finding the Intercepts

To find the x-intercepts, let y be zero and solve the equation for x.

To find the y-intercepts, let x be zero and solve the equation for y

Examples

y = xy = x3 – 4xy2 = x + 4

Symmetry – mirror images

Types of symmetry x – axis if (x,y) then (x,-y)

y – axisIf (x,y) then (-x, y)

OriginIf (x,y) then (-x, -y)

Tests for Symmetry

• If replacing y with –y yields an equivalent equation then symmetric about x-axis

• If replacing x with –x yields an equivalent equation then symmetric about y-axis

• If replacing x with –x and y with –y yields an equivalent equation then symmetric about the origin

Circles

The standard form of the equation of a circle is:

(x – h)2 +(y-k)2 = r2 where (x,y) is a point on the circle, (h,k) is the center and r is the radius

Example

Find the equation of a circle whose center is (-1,2) with point (3,4) on the circle.

Find the radius by using the distance formula

(x +1)2 + (y-2)2 = 20

Review – Section 5

Linear Equations in Two Variables

Slope

is the ratio of vertical change to the horizontal change. The variable m is used to represent slope.

m = change in y-coordinate change in x-coordinate

Or m = rise run

Formula for Slope

m = y2 – y1

x2 – x1

or y x ***this is important

Slope of a Line

Slope Intercept formy= mx + b, where m is the slope, b is y-intercept

Point-Slope formy – y1 = m(x- x1)

Writing Linear Equations in Two Variables

Parallel Lines - are two distinct nonvertical lines having identical slopes; m1 = m2

Perpendicular Lines – are two nonvertical lines whose slopes are the negative reciprocal of each other

Parallel and Perpendicular Lines

The End