Section 8: Linear Equations and Inequalities

8.1 Solving Equations

Definition 13 An equation is a mathematical statement that two expressions are equal.

An equation can be true or false. If an equation contains one or more variables, then it is called an algebraic equation. However, the term equation is commonly used alone to mean algebraic equation. By themselves, many algebraic equations are neither true nor false. That is why we wish to solve them. Solving an equation is the process of determining what numbers, if any, can be substituted for the variable or variables to make the equation true.

Definition 14 Solutions are numbers that make an equation true when substituted for the variable(s) in the equation.

Definition 15 For any particular equation , all of the solutions collectively constitute the solution set.

For some equations, the solution set is obvious. For example most readers could quickiy identify 3 as the solution to 2 +x = 5. To solve more complicated equations, we transform the original equation into a simpler equation. Then the solution set is identified. If needed, the solutions(s) can be substituted into the original equation to check. This transformation process is what many people refer to as "algebraic manipulation" or "symbol manipulation".

Along with the real number properties we've already seen, there are properties of equality that are used in the transformation process. Following the properties are some examples using the properties to solve equations.

1. The Addition Property of Equality: For all real numbers a, b, and c if a = b then a + c = b + c. This says that if I add the same thing to both sides of a true equation, then the equation is still true.
2. The Multiplication Property of Equality: For all real numbers a, b, and c if a = b then ac = bc. This property says that if I multiply both sides of a true equation by the same thing, then it's still a true equation.

Example 27 Using the addition property:


Example 28 Using the multiplication property:


Example 29 Using both the addition and multiplication properties:


Notice that on the left side of the last example, I was strict about the definition of subtraction. Before any other manipulation, the subtraction was replaced addition of the opposite. Later on we will not be so particular about
this issue, but for now it's good to reinforce the definitions. Aso important is that I used the addition property before the multiplication property. When solving equations, you must follow the order of operations backwards. One way to look at the problem is from the viewpoint of what has been done to the variable. The variable will have one or more operations performed on it. You want to "undo" these operations using the Order of Operations in reverse.

Consider the last example above. If you were evaluating the left side for a particular value of x, the order of operations would tell you to multiply by 3 first then add -14 to the result. To solve the equation you undo those steps backwards. First, add the additive inverse of -14 to each side of the equation, then multiply each side of the equation by the multiplicative inverse, or reciprocal, of 3

Next are some slightly more complicated examples that require combining "like" terms. Like terms are any terms that have the same variable and exponent. For example, 3x2 and -5x2 are like terms but 3x and -5x2 are not. Similarly, 3x and -5x are like terms but 3x and -5 are not.

Example 30 In this example the addition or multiplication properties will be used after like terms are combined on both sides of the equation.


Example 31 Here an additional twist is thrown in. Not only are there like terms to combine, but there are terms with the variable on both sides of the = sign. In this case we need to use the addition property to move all variable terms to one side of the equation before we can combine them.


We can summarize the steps to solve these types of equations as follows:

1. Simplify both sides of the equation.
2. Using the addition property, isolate a term with the variable on one side of the equation.
3. Apply the multiplication property to make the coefficient of the variable 1.
(i.e. multiply both sides of the equation by the reciprocal of the numerical coefficient of the variable.) The solution should then be obvious.
4. Cheek the solution by substituting into the original equation.

Let's look at one more example with decimal terms. Some people recommend "clearing" the decimals before solving the equation. That means that you multiply every term in the equation by a number large enough to eliminate any digits past the decimal point. You can certainly do this if you want to, but it's not necessary. You can follow the same steps used before and, if needed, use a a calculator to do the arithmetic.

Example 32


You may have noticed that I did not add the additive inverse of "2" this time. I just subtracted 2 from each side. It's mathematcially equivalent and usually easier to write. As you get more comfortable working with equations you'll make your own decisions about notation issues like this.

Before we move on to equations with parentheses, there is one additional type of equation that you may encounter. Remember the idea of using the order of operations in reverse. Whatever operation that would be performed on the variable first will be the last operations that you" undo". See if you can follow this example:

Example 33


Simlar to the notational shortcut in Example 32, I did not redefine the substraction on the right side of the equation this time. I simply "undid" the subtraction by adding the same value. However, if you are more comfortable replacing the subtractions with addition of opposites, feel free to continue doing so.

8.2 Equations with Parentheses

Often we will encounter equations with one or more expressions in parentheses. In these cases we can use the distributive property of multiplication over addition to eliminate the parentheses. Then we solve the remaining problem just like the previous problems.

Example 34


8.3 Graphing Linear Equations

All the equations we've seen so far are linear equations. Linear has two meanings. The first meaning is that all variables have an exponent of one. In other words, there are no squares or cubes or anything like that. The second meaning is in the context of a graph when there are two variables in the equation.

The first equation example we looked at was x + 8 = -19 and the solution was -27. If this equation had been x + 8 = y, then each possible x would have a different y. This is called an ordered pair and is written (x,y). Every x "pairs up" with a y. Now it's harder to write down the solution set. Unless x has some unusual restriction, it can be any real number. That would give us an infinite number of ordered pairs to write down. So instead trying to write all the solutions we can draw a picture that represents them. This is called a graph of the equation. The graph is placed on something called the Cartesian coordinate plane. Most of us simply think of it as the xy plane:


The x values are plotted along the horizontal axis and the y values are plotted along the vertical axis. Each axis is a number line. On the horizontal, or x, axis positive numbers are to the right and negative numbers are to the left. On the vertical, or y, axis positive numbers are up and negative numbers are down. Thus, the point (3,4) would be 3 spaces right on the x axis and 4 spaces up on the y axis. The point (-2,2) would be 2 spaces left on the x axis and 2 spaces up on the y axis.

Back to the graph of x +8 = y and the second meaning of linear. This is called a linear equation because the graph of its solutions will be a line. Fortunately for us, graphing lines is relatively easy. You need to plot only two points. Then the line that connects those points contains all other solutions.

How do you find two points? Pick any two x values that you like (and that look like they're easy to work with) and put them into the equation. Then solve for y. Let's try 0 and 10 for x. When x =0 we get 0 + 8 = y so y = 8 and our first pair is (0,8). When x = 10 we get 10 + 8= y so y = 18. Thus, the second pair is (10,18). When we plot these points and connect them we get:


As a point of interest, this equation is a function (as are all linear equations). The input variable is x and the output is y. Also, every unique x will give you a unique y. Some of you are familiar with other methods of graphing linear equations (slope-intercept form, point-slope form, etc.). Fee free to use any other methods that you know. However, the "plot two points and connect the dots" method presented here will get you through basic probability and statistics.

8.4 Solving Linear Inequalities

Mathematical statements using <, <, >, and> are called inequalities. Like equations they can be true or false and can use numbers or variables or both. Those with variables would be algebraic inequalities. However, the term inequality alone is commonly used to refer to algebraic inequalities.

Also like equations, any number that makes the inequality true is a solution and all the solutions together constitute the solution set. Unlike equations, where there were usually single solutions, inequalities will usually have many solutions. Before looking at solutions, let's review the meaning of the inequality symbols:

a < b means a is less than b
a < b means a is less than or equal to b
a > b means a is greater than b
a > b means a is greater than or equal to b

By "less" we mean to the left on the number line and by "greater" we mean to the right on the number line. Thus, the following are examples of non-variable inequalities (some are true and some are false):

1 + 8
> 15
-7 + 9
< 10
< -7(-5)
2(-3) + 20
> (-2)(-3)

Obviously, there's no "solving" to be done in the absence of variables but they help demonstrate the meaning of the symbols and the fact that mathematical statements are not necessarily true.

Solving inequalities is only slightly different than solving equations. We will still use properties of multiplication and addition to manipulate the inequality until we get it into a form where the solutions are fairly obvious. The addition property is almost identical to the one for equality:

• For all real numbers a, b, and c, if a > b then a + c > b + c.

The multiplication property, however, has a slight twist. Before stating the property, let's consider two examples.

Example 35: Start with 3 < 8 and mutliple each side by 2 to get 2(3) ?? 2(8) or 6 < 16.

Example 36: Start again with 3 < 8 and mutliple each side by -2 to get -2(3) ?? -2(8) or -6 > -16.

For the statement to remain true in the second example the inequality had to be reversed or "flipped". Thus the multiplication proper for inequalities is:

• For all real numbers a, b, and c with c > 0. If a > b then ac > bc.
• For all real numbers a, b, and c with c < 0. If a > b then ac < bc.

Example 37:


Example 38:


You could convert this answer into a mixed numeral, 6 2/3, but that's of limited value in the digital age. It would probably be recorded as a repeating or rounded decimal (x > -6.66... or x > -6.67).

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