Solving Equations and Inequalities

Introduction

One of the main goals in this course is to help students develop techniques for solving a wide variety of equations and inequalities, which in turn can be used to model real-life situations. This module will focus specifically on linear equations and inequalities.

Linear Equations

An equation can be written in many different ways and still be the same equation. For example:

3x + y = 4 original equation

y = −3x + 4 subtract 3x from both sides

3x − 4 = −y subtract 4 and subtract y from both sides

y − 4 = −3x subtract 4 and subtract 3x from both sides

6x + 2y = 8 multiply both sides by 2

solve for x

These are all different forms of the same equation. It is important to remember that an operation performed on one side of the equal sign must be performed on the other side in order for the expressions to remain equivalent.

Solving Linear Equations

Linear equations that contain symbols of grouping are solved by using the following guidelines:

Remove symbols of grouping from each side by using the distributive property.

Combine like terms.

Isolate the variable using the properties of equality.

Check your solution in the original equation.

Solve Equations Containing Fractions and Decimals

Students are often stumped when dealing with an equation containing a fraction. The fraction can be either a common fraction or a decimal. The unknowns can occupy any position in the equation. If they are part of the fraction, they can be either in the numerator or the denominator. The following are three examples of fractional equations:

Fractional equations are solved using the same approach used for other algebraicequations. However, the initial step is to remove the equation from fractional form. This is doneby determining the lowest common denominator (LCD) for all of the fractions in the equationand then multiplying both sides of the equation by this common denominator. This will clear the equation of fractions. Note how each of the equations noted above is changed.

Multiply both sides by 2: 10x − 1 = 16

Solve:

Multiply both sides by 3: 2x + 6 = 3; 2x = 3 − 6

Solve:

.67x = 3

View a decimal as a fraction; in this case,

Multiply each side by 100: 67x = 300

Solve:

Solve Application Problems

One practical application of Algebra is the use of established formulas to solve common problems. There are various common formulas within geometry, finance, chemistry, and other subjects that everyone uses every day. For example,

Example: Perimeter of a rectangle = 2(length) + 2(width)

Imagine builders are looking at a blueprint of a rectangular room, the length is 1 inch more than 3 times the width, and they need to find the dimensions given that the perimeter is 26 inches. (Note they are looking at a blueprint, so the units are inches.) What do they need to know?

They are looking for the length and width of the rectangle. Since length can be written in terms of width, let w = width.The length is 1 inch more than 3 times the width: 1 + 3w = length. The perimeter of a rectangle = 2(length) + 2(width)

So, 26 = 2(1 + 3w) + 2w

26 = 2 + 6w + 2x

26 = 2 + 8w

24 = 8w

3 = w

So, the width is 3 inches and the length, which is one inch more than three times the width, is 10 inches.

Distance and Midpoint Formulas

Two useful formulas in algebra are the distance formula and the midpoint formula. The distance formula is derived from the Pythagorean Theorem and uses two points plotted in the Cartesian coordinate system to determine the distance between them. The formula is as follows:

Example: Find the distance between the points (2, 5) and (-3, 1).

Solution:

The midpoint formula will find the midpoint between two points plotted on the Cartesian coordinate system. The formula is