# Page 1 Mathematics Study Guide for the TSI

## How to Prepare for the Math Section of the TSI Assessment

### General Information

Colleges and universities today want you to succeed. Most use some sort of placement test before you enroll in college level courses to make sure you have the skills necessary for success.

To find out if you are ready for college math, you will be tested on four basic areas of math on the TSI Assessment. Questions will be drawn from the disciplines of:

• Elementary Algebra and Functions (6)
• Intermediate Algebra and Functions (9)
• Geometry and Measurement (2)
• Data Analysis, Statistics, and Probability (3)

The numbers beside each area above indicate the number of questions about that topic on the placement test. If you demonstrate a need in any of these four areas, according to your placement score (under 350), you will be given a diagnostic test in that area. Each diagnostic test has 12 questions about that area of math, only.

### Elementary Algebra and Functions

Elementary Algebra and Functions covers expressions, equations, functions, and their application in problem-solving.

#### Relations

A relation between two variables, x and y states a relationship between the variables, such as $x \le y$. A relation is true for some x,y pairs such as (3,5) in this example but false for other pairs such as (5,3). In this example the x value must be less than or equal to the y value for the relation to be true.

#### Functions

A function is a specific form of relation where every x value has only one y value that makes the relation true. The previous example, $x \le y$, is not a function because for any x value, there are many y values that are equal to or greater than x. $x=y$, however, is a function because only one value of y can be equal to a given value of x. A function is given in the form $y=f(x)$, where $f(x)$ is an expression with the variable x. It may also be shown as $f(x)=(an \;expression \;with\; x)$.

Evaluating Functions

To evaluate a function for a specific value, substitute the value of the variable into the expression and then perform the indicated operations. The numeric result of the operations is the value of the function for that given value of the variable.

Consider the following function: $f(x) = 3x - 1$, which graphs as:

We can evaluate the function both algebraically and graphically. To evaluate, select an input value, $x = 1$, for example, and substitute it into the function:

Graphically, this means that when $x = 1$, the function evaluates to $f(x) = 2$, which is the point $(1, 2)$. As can be seen from the graph, when $x = 1$, $f(x) = 2$.

Graphing Functions

Pairs of (x,y) values that are true for the function can be graphed with the cartesian coordinate system. The resulting line or curve is called the graph of the function.

Graphing Relations

Relations are graphed in the same way as functions. You may find that the graph is broken in places, or that there are large areas rather than lines, or that there is more than one value of y at any given value of x.

#### Equations, Expressions, and Inequalities

Expressions have one or more variables with operations to be performed, such as $3x + 4$. An equation is a statement that two expressions are equal, such as $y + 1 = x - 2$. An inequality states a relationship where one expression is greater or less than the other. Inequalities can also include the relationships “less than or equal to” and “greater than or equal to.”

Forms of Linear Equations

Linear equations result in a straight line when they are graphed. Linear equations can be written in several forms, and can be converted from one form to another with algebra. One common form is $y=mx + b$, where b is the y-intercept (value of y where the line crosses the y-axis) and m is the slope of the line (how much the y value changes for a given change in x). Other forms reveal other characteristics of the line.

Solving Linear Equations

Solving an equation for a variable means to find an equivalent equation with that variable alone on one side. Do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same quantity. For example, solve $4x + 2y = 8$ for $y$:

subtract 4x from both sides

divide both sides by 2

re-order to put it in the form $y \;=mx \;+\; b$

Solving Linear Inequalities

Solving linear inequalities is very similar to solving equalities. There is one important change. If you multiply or divide by a negative number, the direction of the inequality sign must be reversed.

Consider the following linear inequality: $-3y - 1 \ge 2x + 2$. To solve, begin by isolating y:

$-3y \ge 2x + 3$, and dividing both sides by $-3$, remembering to switch the direction of the inequality:

$y \le -\frac{2}{3}x - 1$, which graphs as:

Graphically, the linear inequality is a bold line (because the inequality is a less than/greater than or equal to, so the points along the line are also included in the solution set), with all values below and along the line shaded- all of these points are shaded because when plugged into the original inequality, they yield true statements; consequently, the make up the solution set.

Linear Systems of Equations

A set of linear equations where the number of equations matches the number of variables is a linear system of equations. A value of (x,y) that works in both equations is the solution of the system. This is the spot where the lines intersect. If the lines are parallel, there are no solutions.

Consider the following system of linear equations:

$-x + y = 1$, and
$2x + y = 4$, which graphs as:

The point, $(1,2)$, marked in red, indicates the solution to the system of equations. It is the point that lies along both lines, and marks their intersection.

Here are two methods for solving systems of equations:

Substitution—The system $x + y = 5$ and $x - 2y = -2$ will be solved by substitution:

This will be substituted into the second equation for x:

Now, solve for y:

Now, substitute this value back into the first equation and solve for x:

The solution of the system is $x = 4$, $y = 1$. These values work in both equations.

Elimination—In elimination, the equations are added or subtracted to eliminate one of the the variables. An equation can be multiplied as well. The goal is to get coefficients of the same variable to be the same or opposite in both equations.

For the same system, multiply the first equation by 2 to get:

The other equation is $x - 2y = 2$.
Notice that the coefficient of y is opposite. If the equations are added, y will be eliminated:

Substitute this back into either equation to get $y = 1$.

Algebraic Expressions

An algebraic expression looks like an equation but there is no equal sign. It is essentially a recipe describing what to do with a variable to get the desired result. For example, the expression $5x - 3$ means to multiply the value of x by $3$ and then subtract $3$.

Word Problems

Many real-life situations can be described with linear models. Often, the information given in a problem can be used to form an equation that describes the situation. This is the model. Usually, the equation can be solved to find the required solution. For example, suppose a car is traveling at a rate of $50$ miles per hour. How long does it take to travel $120$ miles?

If $y$ is miles traveled, and $x$ is the hours traveled, then the model is $y = 50x$.
In this case we know that $y$ is 120 miles, so $120 = 50x$.
To solve for $x$, we have to divide both sides by 50, so $x = 120 \div 50$, or $2.4$ hours.