Learning Resource Services CLAST Review Algebra Learning Resource Services

Real Numbers Real numbers are: Rational numbers Irrational numbers either terminating (1.0, -5.2) or repeating decimals (3.17555) Irrational numbers non-terminating and non-repeating decimals (, ) All the rules that apply to rational numbers in arithmetic also apply to irrational numbers. CLAST Algebra

Addition of Irrational Numbers To add or subtract irrational numbers, first simplify each irrational term and then combine like terms. Example: 2 + 7 -  + 3 + 4 all terms are simplified combine like terms 2 -  + 4 + 7 + 3 = 5 + 10 Example: Simplify terms Combine like terms CLAST Algebra

Multiplication of Irrational Numbers Example: Combine radicands under one radical sign Factor Simplify CLAST Algebra

Division of Irrational Numbers Example: Combine radicands under one radical sign Factor Simplify CLAST Algebra

Rationalize the Denominator Do not leave a radical in the denominator Example: Multiply the radical denominator by itself to form a perfect square. Multiply the numerator by the same term, to keep the value of the fraction (this is actually multiplying by 1) Simplify CLAST Algebra

apply the Order of Operations Agreement By convention, there is a specific order of operations. This mnemonic will help you to remember the order: Please Excuse My Dear Aunt Sally Parentheses solve the terms inside parentheses first Exponents evaluate exponents Multiply and/or Divide from left to right Add and/or Subtract from left to right CLAST Algebra

use Properties of Operations Associative (grouping) a + (b + c) = (a + b) + c a(bc) = (ab)c Commutative (order) a + b = b + a ab = ba Distributive a(b + c) = ab + ac Inverse a + (-a) = 0 a ( ) = 1 if a 0 Identity a + 0 = a CLAST Algebra

use Scientific Notation Scientific notation always appears as a factor (one non-zero integer before the decimal portion) multiplied by a power of 10 Decimal Form Scientific Notation 0.0034 3.4 x 10-3 5.46 5.46 x 100 21,500 2.15 x 104 CLAST Algebra

Determine if a Number is a Solution to an Inequality Is x = 5 a solution to the inequality (x - 5)(x + 4)  0 ? Substitute 5 for x and simplify each side (x - 5)(x + 4)  0 (5 - 5)(5 + 4) 0 (0) (9) 0 0 0 zero is equal to zero x = 5 is a solution CLAST Algebra

Properties of Equality and Inequality i. If a = b, then a + c = b + c ii. If a > b, then a + c > b + c iii. If a = b, then ac = bc iv. For c > 0, if a > b, then ac > bc v. For c < 0, if a > b, then ac < bc vi. If a > b and b > c, then a > c CLAST Algebra

Use Properties to Identify Equivalent Equations and Inequalities Example: 2 x + 1 = 11 use property i. 2 x + 1 + (-1) = 11 + (-1) 2 x = 10 Example: , given y > 0 use property iv. CLAST Algebra

Solve Linear Equations and Inequalities To solve an equation or inequality, find its solution. A solution is a value which makes the equation or inequality true when the value is substituted for the variable. Example: Determine whether x = -1 is a solution of the equation x + 5 = 4. -1 + 5 = 4 Substitute -1 for x. 4 = 4 A true statement Therefore, x = -1 is a solution CLAST Algebra

Steps to Find the Solution for an Equation Remove grouping symbols (parentheses) using the distributive property. Use properties to move all variable terms to one side of the equation. Use properties to move all non-variable terms to the other side of the equation. Simplify each side of the equation. Multiply both sides of the equation by the reciprocal of the numerical coefficient of the variable term. A number multiplied by its reciprocal equals 1: CLAST Algebra

Use Algebraic Formulas An algebraic formula establishes a relationship between two or more variables. Example: I = P • R • T Interest = Principle • Rate • Time CLAST Algebra

Steps to Evaluating Formulas Identify any given values Substitute given values into the formula in place of the variables Solve the formula for the remaining variable Label your answer with the appropriate unit CLAST Algebra

Functions A function establishes a special kind of relationship between two sets of variables. One set is called the independent variable (usually the x values). All of the possible x values are called the domain of the function. The other set is called the dependent variable (usually the y values). All of the possible y values are called the range of the function. A function relates domain to range Each value of the domain corresponds to one and only one value of the range. That is, each value assumed by x determines one and only one value of y. CLAST Algebra

Examples of Functions A set of ordered pairs (x and y values) {(1, 2), (2, 4), (3, 6)} An equation y = 2x An equation written in functional notation f(x) = 2x CLAST Algebra

Visual Examples of Functions Remember, a function is defined so that each value of the domain corresponds to one and only one value of the range. One method to visually determine whether a graph represents a function is to apply the “vertical line” test. If a vertical line passed over the graph touches more than one point at a time, the graph does not represent a function. . not a function . function CLAST Algebra

Find Values of Functions Functional notation states the value which is to be substituted for x into an equation in order to determine the corresponding value for y: Example: Given f(x) = 2x - 3, find f(2) Substitute the value 2 for x in the equation f(2) = 2(2) - 3 = 4 - 3 = 1 CLAST Algebra

Find Factors of Quadratic Expressions A quadratic expression contains one term in which the variable is squared. Example: 2x2 + 5x + 2 The factors of a quadratic expression are two linear expressions which multiply together to form the original quadratic expression. (2x + 1)(x + 2) = 2x2 + 4x + 1x + 2 = 2x2 + 5x + 2 Outside terms Last terms First terms Inside terms CLAST Algebra

Find Factors of Quadratic Expressions page 2 Example: (3x - 2)(x + 4) = 3x2 + 12x - 2x - 8 = 3x2 + 10x - 8 The two linear factors are determined by the FOIL technique: The product of the First term of each factor is the first term of the quadratic The products of the Outside terms and the Inside terms of the factors, added together, is the middle term of the quadratic The product of the Last term of each factor is the last term of the quadratic Outside terms Last terms First terms Inside terms CLAST Algebra

The FOIL Technique Example: Determine the linear factors of the quadratic expression 4x2 - 9x + 2 Step 1: Determine all possible pairs of factors of the First term 4x and x 2x and 2x -4x and -x -2x and -2x Step 2: Determine all possible pairs of factors of the Last term 2 and 1 -2 and -1 CLAST Algebra

The FOIL Technique page 2 4x2 - 9x + 2 Step 3: Systematically test one pair of factors from step 1 with one pair of factors from step 2 until the middle term of the quadratic is formed as the sum of the products of the Outside and Inside terms. (4x + 2)(x + 1) 4x + 2x = 6x No (4x + 1)(x + 2) 8x + 1x = 9x No (4x - 2)(x - 1) -4x - 2x = -6x No (-4x - 2)(-x - 1) 4x + 2x = 6x No (4x - 1)(x - 2) -8x - 1x = -9x Yes through trial and error, 4x2 - 9x + 2 = (4x - 1)(x - 2) Outside + Inside CLAST Algebra

Find Solutions to Quadratic Equations The solution to a quadratic equation of the form ax2 + bx + c = 0 can be found by factoring set each factor equal to zero, then solve each of the resulting linear equations substituting the values for a, b, and c into the quadratic formula: CLAST Algebra

A System of Two Linear Equations A system consists of two or more linear equations, each of the form ax + by = c. Each equation has an infinite number of ordered pair (x, y) solutions. A system represents two lines on the coordinate plane. The solution of the system can be one ordered pair (the two lines intersect) no ordered pairs (the two lines do not intersect), or an infinite number of ordered pairs (the two lines are the same) CLAST Algebra

Solve a System of Two Linear Equations in Two Unknowns: the Addition Method Write each equation in standard form. Determine which coefficients of the x and y terms will be the easiest to make into opposite values. If necessary, multiply one or both of the equations by a constant to make the chosen set of coefficients opposites. Add these two new equations. The result is a single equation with one variable. Solve for this variable. Substitute this value into either of the original equations and solve for the other variable. Check your answer. CLAST Algebra

Example of the Addition Method Solve the system of equations 2x - y = 4 3x - 2y = 2 Step 1: The equations are already written in standard form. The y terms will be easier to make opposites. Step 2: Multiply the first equation by -2 Step 3: The new system of equations is -4x + 2y = -8 3x - 2y = 2 CLAST Algebra

Addition Method page 2 Add the two equations -4x + 2y = -8 Multiply this equation by -1 to determine that x = 6. Step 4: Substitute x = 6 into the original form of the first equation to solve for y. 2(6) - y = 4 The solution is the ordered pair (6, 8) 12 - y = 4 Check in first equation: 2(6) - 8 = 4 - y = -8 12 - 8 = 4  y = 8 Check in second equation: 3(6) - 2(8) = 2 18 - 16 = 2  CLAST Algebra

Identify Specified Regions of the Coordinate Plane A coordinate plane is where ordered pairs are graphed in relation to a horizontal axis (x) and a vertical axis (y). A linear inequality, when graphed, produces not just one solution, but a region of solutions. Example: 2x - y < 6 Step 1: Sketch the boundary line 2x - y = 6 x y 0 -6 3 0 y . x boundary line < dashed < solid > dashed > solid . CLAST Algebra

Regions of the Coordinate Plane page 2 Step 2: Shade the region that contains the solutions Choose a test point not on the boundary line and substitute this point into the original inequality. Let the test point be (0, 0) 2x - y < 6 2(0) - 0 < 6 0 < 6 a true statement If true, shade the region that contains the test point If false, shade the region that does not contain the test point y . x • . CLAST Algebra

Problems Involving the Structure and Logic of Algebra An algebraic relationship between two or more quantities may sometimes be stated as a word problem. The words need to be translated into math symbols so that a solution can be determined. Choose a variable to represent one quantity Express all other quantities in terms of this variable Express the relationship symbolically as an equation or inequality CLAST Algebra

Solve Problems Involving the Structure and Logic of Algebra Example: Write the equation needed to find a number if that number, increased by five times that number, is 48. Let n represent the number 5n represents five times the number + represents “increased by” = represents “is” The equation needed is n + 5n = 48 CLAST Algebra

Identify Statements of Proportionality and Variation Two quantities, represented by the variables x and y, can be related in many ways. Two special relationships are that x and y vary directly: x produces y Direct Variation equation y = kx or Ratio x and y vary inversely: x produces y Inverse Variation equation xy = k or Ratio k is the constant of proportionality CLAST Algebra

Example of Direct Variation When traveling, distance equals rate times time, or d = rt Distance increases if rate or time increases Distance decreases if rate or time decreases Example: If r = 10 miles/hour and t = 1 hour, then d = 10m/hr x 1 hr = 10 miles If r increases to 20 miles/hour and t remains 1 hour, then d = 20m/hr x 1 hr = 20 miles (an increase) CLAST Algebra

Example of Inverse Variation When traveling, distance equals rate times time, or d = rt . This formula can be solved for r, as r = If distance remains the same, rate will change inversely to time Example: If d = 10 miles and t = 1 hour, then r = 10 miles/1 hour = 10 miles/hour If d remains 10 miles and t increases to 2 hours, then r = 10 miles/2 hours = 5 miles/hour CLAST Algebra

Algebraic Word Problems with Variables The algebraic solution to a word problem includes the following steps: Read the problem carefully Organize the problem on paper Substitute values into a given formula. If no formula is given, identify the relationship between the various quantities and express that relationship as an equation. Solve for the unknown variable Check the solution CLAST Algebra