HA - Linear Equations in One Variable 2 Lesson

 

Linear Equations in One Variable

Using One Inequality

The previous lesson provided a review of the properties of addition, subtraction, multiplication, and division plus parentheses and common denominators. With this background, one change is made to explain inequalities.  Instead of only equals, =, the use of >, ≥ , <, ≤, ≠ will be used in this section.

Inequality - A mathematical statement indicating that two quantities are not strictly equal, but may have a relationship with or without equality. A mathematical inequality statement will use the symbols:

 

≤	Less than or equal to, indicates that the math expression on the left is smaller or equal to the math expression on the right;  5 ≤ 6 + 2 or 5 ≤ 6 -1.
≥	Greater than or equal to, indicates that the math expression on the left is greater or equal to the math expression on the right: 10 ≥   6 - 4 or 10 ≥ 3 + 7.
<	Less than, indicates that the math expression on the left is smaller than the math expression on the right;  5 < 6.
>	Greater than, indicates that the math expression on the left is greater than the math expression on the right;  7 > 2.
≠	Not equal to, indicates that two numbers or expressions are not equal; 8 ≠ 9.

First, let's understand the meaning of an inequality.

x ≥ 2

This is read, "x is greater than or equal to 2". This means that x could stand for the number 2 or for any number greater than 2, not just whole numbers, but numbers that could be fractions. Some possible values for x: 2,  = 2.5, 100, or 2 million.

x > 2

This is read, "x is greater than 2". This means that x could stand for any number greater than 2, not just whole numbers, but numbers that could be fractions. Some possible values for x:  = 3.75, 100, or 2 million, but x could not be the number 2.

x ≤ 2

This is read, "x is less than or equal to 2". This means that x could stand for the number 2 or for any number less than 2, not just whole numbers, but numbers that could be fractions.

Some possible values for x: 2,  = 1.5, 1, 0, - = - -0.5, -16 or negative one million.

x < 2

This is read, x is less than 2. This means that x could stand for any number greater than 2, not just whole numbers, but numbers that could be fractions.  So x might be  = 1.5, 1,  = 0.75, 0, -  = -0.5, -16 or negative one million.

x ≠ 2

This is read, "x is not equal to 2".  This means that x could stand for the any number other than 2, not just whole numbers, but numbers that could be fractions.  

Some possible values for x: 100, 17,  = 1.5, 1, 0, - = -.8, -16 or negative one million, but x is not a placeholder for 2.

All of the rules that were presented for one variable equation solving are the same.

General Equations Rule

Going from equality to inequality this rule remains the same.

 

Inequality Rule is Added

This rule only applies to inequality equations using >, ≥, <, ≤.

 

Just like the one variable equality equations, when asked to find the solution or answer, the inequality will be solved when the variable is on one side of the inequality and the rest of the numbers are combined on the other side of the equality.

Check out the examples below. 

The next video is a demonstration of the basic inequalities using the techniques that were learned in Lesson 1.  Remember that all the rules for addition, subtraction, multiplication, and division remain the same, but we add one rule.  If the inequality is multiplied or divided by a negative number, the inequality sign is flipped.

 

Note the two methods used to solve the equation in the video.  For the first method,  the 2 was moved by dividing each side by 2.  For the second method, the distributive property was used first. Each method resulted in the same answer. When solving problems, always look for the easiest method to use. Do not create fractions during the problem solving unless absolutely necessary. Most problems can be solved by clearing the fractions and working for the solution. The equation may still result in a fraction by dividing in the last step.  

The video below provides examples of the problems that clear the fraction at the beginning of the problem and return the fraction with a final division.

 

Another inequality is the symbol, not equals, ≠.  This one is straight forward.  The not equals symbol explains that one side of the equation is not equal to the other.  

Check out the example below. 

 

Check your inequality knowledge by solving for the variable and checking your answers. 

One Variable Equation with Two Inequality Signs (Compound Inequality)

Sometimes the solution to the inequality is between two numbers or excluding a middle section of numbers. These situations are called compound inequalities.

Check out the example below. 

Absolute Value Inequalities

Remember that Absolute Value means that the answer will always be positive even if a negative number is in the absolute value.  |-7| = 7 and |7| = 7.

Example:  The basic example of absolute value.

|x| ≥ 10

With absolute value there are two cases, a positive case and a negative case because |-10| = 10 and |10| = 10

Given the inequality |x| < a (or |x| ≤ a), the solution is always of the form –a < x < a (or -a ≤ x ≤ a). Notice that inequalities in this form are read using the word "and." -a < x < a is read as "x is greater than -a and x is less than a."

Given the inequality |x| > a (or |x| ≥ a), the solution always starts by splitting the inequality into two pieces: x < –a or x > a (or x ≤ -a or x ≥ a). Notice that inequalities in this form are read using the word "or."

Case 1:  Positive Case:  

x ≥ 10

Case 2: Negative Case:

-x ≥ 10 Divide by-1 to have a positive x.

x ≤ -10 Division by - 1 flips the inequality sign

So there are two sections of numbers that solve the inequality, all numbers that are less than or equal to -10 and all numbers that are greater than or equal to 10:   x ≤ -10 or x ≥ 10.

Watch the video to examine more concerning Absolute Value and Inequalities. On the video there is also an example of how to find the Absolute Value equation when the answers are provided.

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