Midterm Review Algebra 2

Types of Inequalities Linear: Isolate the variable Intersection or overlap when the inequalities are joined by “and” Union, include both sets when the inequalities are joined by “or” Absolute Value: Two inequalities using “and” or “or”

Linear equation 3(x + 5) – 7 = 4x – (x+3) 3x + 15 – 7 = 4x – x – 3 8 = – 3 No Solution

Absolute Value Equation 3 |2x – 5| +7 = 28 3 |2x – 5| = 21 |2x – 5| = 7 Interpret: 2x – 5 = 7 or 2x – 5 = – 7 2x = 12 or 2x = – 2 x=6 or x = – 1 {– 1, 6}

Polynomial Equation 2x2 + x = 10 2x2 + x – 10 = 0 (2x + 5)(x – 2) = 0 2x + 5 = 0 or x – 2 = 0 2x = – 5 or x = 2 x = – 5/2 or x = 2 {– 5/2, 2}

Linear Inequalities 3(x + 2) < 2(4x – 7) distribute 3x + 6 < 8x –14 subtract 8x and 6 -5x < -20 divide by –5 and flip x > 4 solution (4, ) interval notation and graph 0 4

Absolute Value Inequality “and” |3x – 5| 10 3x – 5 10 and 3x – 5 -10 Can be written –10 3x – 5 10 -5 3x 15 -5/3 x 5 [-5/3, 5] -5/3 0 5

Absolute Value Inequality “or” |2x – 5| > 3 2x – 5 > 3 or 2x – 5 < -3 2x > 8 or 2x < 2 x > 4 or x < 1 Interval notation: 0 1 4

Interval Notation and Graphing < or > don’t include the value, parenthesis use open circle on graph or include the value, square bracket use closed circle on graph Always use parentheses with infinity -3 5

Lines General linear equation: ax + by = c Slope-intercept equation: y = mx + b Point-slope equation: y-y1= m(x – x1) Slope: Parallel lines have the same slope. Perpendicular lines have slopes that are opposite reciprocals. Vertical lines have no slope.

Functions A rule that assigns to each element of the domain a unique element of the range. A set of ordered pairs (x,y) such that each x corresponds to one and only one y. The graph of a function intersects any vertical line in at most one point. f(x) notation.

Domain The set of all inputs to the function. Since this variable is usually called x, the domain is the set of all values for x.

Range The set of all outputs of the function. Since this variable is usually called y, the range is the set of all values for y.

f(x) Notation f(x) = 3x – 7 x is the input and f(x) is the output. 3x – 7 is the rule. Input 2 f(2) = 3(2) – 7 = –1 is output. (2, –1) corresponds to a point on the graph of this linear function. More

Graph of a Function The set of all points corresponding to the ordered pairs (x,f(x)) is the graph of the function, i.e. let y = f(x). If a vertical line intersects the graph in more than one point then it is not the graph of function.

More Function Notation Let f(x) = x2 – 2x f( ) = ( )2 – 2( ) Find f(x – 3) f(x – 3) = (x – 3)2 – 2(x – 3) = x2 – 6x + 9 – 2x + 6 = x2 – 8x + 15