Chapter 7, Section 1.

To solve a linear equation in one variable:

1.  If there are parenthesis but no fractions and/or decimals, go to 1c:
    1a.  If there are fractions, multiply both sides of the equation by the LCD to clear the fractions, then go to 1c.
    1b.  If there are decimals, multiply both sides of the equation by a power of 10 to clear the decimals, then go to 1c.
    1c.  Use the distributive property to clear all parenthesis.

  2.  Combine like terms on either side of the equation.

  3.  Use the addition property to get all variable terms on one side of the equation and all constant terms on the other side of the equation.

  4.  Use the multiplication property of equality to get x by itself on one side of the equation and the constant term on the other side of the equation.

  5.  Check your answer in the original equation.

  Remember, if the variable “disappears” and you get
   1.  a contradiction such  as 3 = 7, the equation has “no solution”.
   2.  an identity such as 4=4, the equation has all real numbers as its solution.

 Next is an example.  I will leave the check to you!
Example  (has every step-not like typical homework).

Chapter 7, Section 2.

  To solve a linear equation in one variable that also has absolute value symbols:

  First, isolate the absolute value on one side of the equation.  If there are two sets of absolute value symbols, one set should be on either side of the equal sign.

  Next, rewrite the absolute value equation as two separate equations as follows:
   ---Drop the absolute value symbols but leave everything else as is.
   ---Then, rewrite the left side of the equation to equal the opposite of the right            side of the equation.
    --Follow the process for solving linear equations.

  ALL ANSWERS MUST BE CHECKED IN THE ORIGINAL PROBLEM!

  Examples.

Chapter 7, Section 3.

Review:  To solve a linear inequality in one variable:

Use the same process steps as in Chapter 7, Section 1, with the following exception:

If a process solving step involves multiplication by or division by a negative number, then reverse the direction of the inequality symbol.

Also, most of these problems must be graphed.  If the inequality symbol does not have an “equal to bar”, use an open circle on the graph.  If the inequality symbol has an “equal to bar”, use a closed circle on the graph. 

If asked to also provide interval notation, then the open circle, along with infinity and negative infinity symbols uses parenthesis in the interval.  The closed circle uses brackets. 

In this section, the concept of “OR” and “AND” is introduced.  If the connecting word is OR, graph all points that satisfy either of the given inequalities.  If the connecting word is AND, graph all points that satisfy both of the given inequalities (or that are common to both graphs).

If your graph is more than one region on the number line, join the intervals using the “union of two or more sets” notation, a capital U.

NOTE:  There is an alternate method to write the AND statements which contain two inequality symbols.  Remember to keep the variable in the middle portion and undo to all three parts in order to isolate the variable. 

Use your book to read examples of these processes with examples 2, 4 and 5, pages 420-422.  I can’t build graphs on my word processor!

Chapter 7, Section 4.

To solve an absolute value inequality in one variable:

Remember, as with the equation, isolate the absolute value symbol on the left side of the inequality FIRST!

Then,
   --rewrite the absolute value inequality with no absolute value symbols,
   --choose the appropriate connecting word (and if the symbol is “less thAND”,        or if the symbol is “greatOR than”)
    --rewrite the left side of the inequality, reverse direction of the inequality symbol       and write the opposite of the right side of the inequality. 
    --Isolate the variable using prior lessons (see Chapter 7, section 1). 

Example 6, page 428 will illustrate this process.  Remember, we did our AND statements differently than the book does, so examples 1,2,5 “look differently” than those done in class.  Let’s see example 2 our way: