Algebra for Nursing School Entrance Exam Study Guide (page 3)
The practice quiz for this study guide can be found at:
Popular topics for algebra questions on nursing school exams include:
- Solving equations
- Positive and negative numbers
- Algebraic expressions
What Is Algebra?
Algebra is a way to express and solve problems using numbers and symbols. These symbols, called unknowns or variables, are letters of the alphabet that are used to represent numbers.
For example, let's say you are asked to find out what number, when added to 3, gives you a total of 5.Using algebra, you could express the problem as x + 3 = 5. The variable x represents the number you are trying to find.
Here's another example, but this one uses only variables. To find the distance traveled, multiply the rate of travel (speed) by the amount of time traveled: d = r × t. The variable d stands for distance, r stands for rate, and t stands for time.
In algebra, the variables may take on different values. In other words, they vary, and that's why they're called variables.
Algebra uses the same operations as arithmetic: addition, subtraction, multiplication, and division. In arithmetic, we might say 3 + 4 = 7, while in algebra, we would talk about two numbers whose values we don't know that add up to 7, or x + y = 7. Here's how each operation translates to algebra:
An equation is a mathematical sentence stating that two quantities are equal. For example:
2x = 10
x + 5 = 8
The idea is to find a replacement for the unknown that will make the sentence true. That's called solving the equation. Thus, in the first example, x = 5 because 2 × 5 = 10. In the second example, x = 3 because 3 + 5 = 8.
Sometimes you can solve an equation by inspection, as with the above examples. Other equations may be more complicated and require a step-by-step solution, for example:
The general approach is to consider an equation like a balance scale, with both sides equally balanced. Essentially, whatever you do to one side, you must also do to the other side to maintain the balance. Thus, if you were to add 2 to the left side, you would also have to add 2 to the right side.
Let's apply this balance concept to our previous complicated equation. We want to solve for n, which means we must somehow rearrange it so the n is isolated on one side of the equation. Its value will then be on the other side. Looking at the equation, you can see that n has been increased by 2 and then divided by 4 and ultimately added to 1. Therefore, we will undo these operations to isolate n.
Notice that each operation in the original equation was undone by using its inverse operation. That is, addition was undone by subtraction, and division was undone by multiplication. In general, each operation can be undone by its inverse.
After you solve an equation, check your work by plugging the answer back into the original equation to make sure it balances. Let's see what happens when we plug 6 in for n:
Positive and Negative Numbers
Positive and negative numbers, also known as signed numbers, are best shown as points along the number line:
Numbers to the left of (smaller than) 0 are negative and those to the right are positive. Zero is neither negative nor positive. If a number is written without a sign, it is assumed to be positive. Notice that when you are on the negative side of the number line, bigger numbers have smaller values. For example, –5 is less than –2.You come into contact with negative numbers more often than you might think; for example, very cold temperatures are recorded as negative numbers.
As you move to the right along the number line, the numbers get larger. Mathematically, to indicate that one number, say 4, is greater than another number, say –2, the greater than sign (>) is used:
4 > –2
On the other hand, to say that –2 is less than 4, we use the less than sign (<):
–2 < 4
Arithmetic with Positive and Negative Numbers
The table on the next page illustrates the rules for doing arithmetic with signed numbers. Notice that when a negative number follows an operation, it is often enclosed in parentheses to avoid confusion.
When more than one arithmetic operation appears, you must know the correct sequence in which to perform the operations. For example, do you know what to do first when calculating 2 + 3 × 4? You're right if you said, "Multiply first." The correct answer is 14. If you add first, you'll get the wrong answer of 20! The correct sequence of operations is:
Even when signed numbers appear in an equation, the step-by-step process works exactly as it does for positive numbers. You just have to remember the arithmetic rules for negative numbers. For example, let's solve –14x + 2 = –5.
An algebraic expression is a group of numbers, unknowns, and arithmetic operations, like 3x – 2y. This one may be translated as "3 times some number minus 2 times another number." To evaluate an algebraic expression, replace each variable with its value. For example, if x = 5 and y = 4, we would evaluate 3x – 2y as follows:
- 3(5) – 2(4) = 15 – 8 = 7
Squares and Square Roots
It's not uncommon to see squares and square roots on standardized math tests, especially on questions that involve right triangles.
To find the square of a number, multiply that number by itself. For example, the square of 4 is 16, because 4 × 4 = 16.Mathematically, this is expressed as:
42 = 16
4 squared equals 16.
To find the square root of a number, ask yourself, "What number times itself equals the given number?" For example, the square root of 16 is 4 because 4 × 4 = 16.Mathematically, this is expressed as:
The square root of 16 is 4.
Because certain squares and square roots tend to appear more often than others on standardized tests, the best course is to memorize the most common ones.
How to Solve an Equation
- Example: 5 – 2(3x + 1) = 7x – 8
- Remove parentheses by distributing the value outside to both values within:
- 5 – 6x – 2 = 7x – 8
- If there are like terms on the same side of the equal sign, combine them:
- 3 – 6x = 7x – 8
- Decide where you want all of the x terms. Put all the x terms on one side by addition and subtraction:
- Get all the constants on the other side by addition and subtraction:
- Isolate the x by performing the opposite operation:
The practice quiz for this study guide can be found at:
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