Archive for February, 2013

Solving Equations in One Variable in Beginning Algebra Courses

February 26, 2013

Fundamental Facts about Equations

Definition: Two equations are equivalent if they have the same solution set.

Definition: A simplest equation is an equation which has a single variable on one side of the equal sign and a single number on the other side.

Zero Factor Property: If a and b are real numbers and ab = 0, then a = 0 or b = 0.

Quadratic Formula: The solutions of a quadratic equation ax2 + bx + c = 0 are given by

 x=frac{-bpm sqrt{{{b}^{2}}-4ac}}{2a}

Four Fundamental Properties of Equations:
1. If any expression is added to both sides of an equation the resulting equation is equivalent to the original equation.

2. If both sides of an equation are multiplied by the same non-zero real number, the resulting equation is equivalent to the original equation.

3. When both sides of an equation are squared the two equations may not be equivalent but the solution set of the resulting equation contains the solution set of the original equation.

4. When both sides of an equation are multiplied by an expression containing a variable the two equations may not be equivalent but the solution set of the resulting equation contains the solution set of the original equation.

Solution Sets of Equations

Armed with these facts the student should think of equation solving in the following simple, general, and abstract terms.

If the equation is linear, the first two fundamental properties of equations are used to generate a sequence of equations each equivalent to the previous one until a simplest equation is obtained.  The solution set for this simplest equation is obviously the solution set for the original equation.

If the equation is quadratic, the first two fundamental properties of equations are used to rewrite (if necessary) the equation in standard form.  Either factoring combined with The Zero Factor Property (followed by applications of the first two fundamental properties of equations as needed) or The Quadratic Formula is used to generate two simplest equations, conjoined with the word or, equivalent to the original quadratic equation, so that the union of the two solution sets is the solution set of the original quadratic equation.

If the equation contains radicals, the first two fundamental properties of equations are used to rewrite the equation (if needed) so that squaring both sides of the equation will eliminate the radical. It may be necessary to repeat this.  When no radicals remain, the resulting equation is solved using one or more of the above strategies.  The solution set so obtained contains the solution set of the original equation.  Each of these possible solutions must be tested in the original equation to determine the solution set for the original equation.

If the equation is a rational equation, both sides of the equation are multiplied by the least common denominator.  If the resulting equation is linear or quadratic, it may be solved using one or more of the above strategies. The solution set so obtained contains the solution set of the original equation.  Each of these possible solutions must be tested in the original equation to determine the solution set for the original equation.

In fact these last two strategies may be combined as soon as the student recognizes that each of

  • multiplying both sides of an equation by an expression containing a variable, or
  • squaring both sides of an equation

generates an equation which may not be equivalent to the original equation but whose solution set contains the solution set of the original equation.

Graphs of Equations

The graph of a linear equation in one variable will be a single dot on the Real Number line.

The graph of a quadratic equation in one variable will be one of the following:

  • A single dot on the Real Number line
  • Two isolated dots on the Real Number line
  • No graph (if the solution set contains complex numbers)

A rational equation in one variable has no meaning for those values of the variable which create a zero in a denominator (division by zero is undefined).  Consequently the graph of a rational equation in one variable appears (is sketched) on a Real Number line with those numbers removed.  The graph of a rational equation in one variable is a collection of isolated dots on a Real Number line which has been modified by removing those numbers which create a zero in a denominator.

An equation in one variable which contains radicals is undefined in the Real Number system for those numbers which create a negative radicand (the square root of a negative number is not a Real Number).  Consequently the graph of an equation in one variable containing radicals appears (is sketched) on a Real Number line with those numbers removed.  The graph of an equation in one variable containing a radical is a collection of isolated dots on a Real Number line which has been modified by removing those numbers which create a negative radicand.

The Law of Trichotomy

The Law of Trichotomy is a property of the Real Numbers which is seemingly quite obvious and at the same time is quite powerful.

Law of Trichotomy: If a and b are real numbers, then one and only one of the following is true:

  1. a < b
  2. a = b
  3. a > b

This property is obvious in the sense that if you write down two Real Numbers, then clearly one and only one of the following is true:

  1. The first is less than the second
  2. The first is equal to the second
  3. The first is greater than the second

This property is powerful in the sense that if a real number (for which the equation makes sense) is substituted into an equation then one and only one of the following is true:

  1. The left side is less than the right side
  2. The left side is equal to the right side
  3. The left side is greater than the right side

Equations and Corresponding Inequalities

Recognize that for every equation, two inequalities are easily generated by replacing the = symbol with the symbols < and >.  The Law of Trichotomy literally begs us to consider all three (corresponding less than inequality, equation, corresponding greater than inequality) each and every time we consider any one of them.  Such joint consideration is made simple by considering their graphs.

The graph of an equation in one variable consists of a collection of isolated dots on a Real Number line (possibly modified by deleting some numbers).  Those dots (points on the graph) together with deleted points (if any) divide the number line into a collection of rays and intervals. Each ray or interval is a part of the graph of exactly one of the inequalities and every number in that ray or interval is a solution of that inequality.  Therefore the solution sets for each of the inequalities may be determined by testing a single number from each ray and each interval in either one of the inequalities.  The graph of the equation forms the boundary for the graphs of the corresponding inequalities.  It is therefore called the “boundary equation”.

When some numbers are not part of the domain, they are also boundary points.

Illustrations

image6  image5
 image4
 image3

Equations and Corresponding Inequalities Involving Absolute Value

Definition: The absolute value of a number is defined by:

absolute_value_def

For the sake of simplicity, the following discussion of absolute value equations and inequalities will be restricted to absolute values of linear expressions.  Much of this is true more generally, but this discussion is explicitly for Intermediate and College Algebra classes.  Extensions can easily be made in other courses when necessary.

For this discussion of equations and inequalities involving absolute values we restrict attention to equations and inequalities of the form

|M| < k                         |M| = k                         |M| > k

where M is some linear expression of the form ax + b and k is a real number.

The Law of Trichotomy dictates that k < 0, k = 0, or k > 0.  The three possibilities for k combined with the boundary equation and the two inequalities forces us to consider nine cases.  Fortunately most of them are trivial and can be dealt with quite easily.  We will separately consider the three possibilities (k = 0, k < 0, and k > 0) for the number k.

Remember that the absolute value of an expression is nonnegative and is zero if and only if the expression inside the absolute value sign is zero.

If k < 0, then |ax + b| > k is true for all real numbers, and the solution set for each of |ax + b| = k and |ax + b| < k is the empty set.

If k = 0, then the solution set for |ax + b| < k is the empty set, the equation |ax + b| = k is equivalent to the linear equation ax + b = 0 whose solution set is left{ -frac{b}{a} right}. From the Law of Trichotomy it follows that in this case the solution set for |ax + b| > k is the set of all real numbers except -frac{b}{a} .

If k > 0, the equation and corresponding inequalities are a bit more interesting.

In this case the inequality |ax + b| < k is equivalent to the compound compact inequality -k < ax + b < k whose solution set is an interval (m, n) of the real numbers.  The solution set of the boundary equation |ax + b| = k is the set {m, n}.  From the Law of Trichotomy, it follows that the solution set for the inequality |ax + b| > k is the union union_of_raysof the two rays (-∞, m) and (n, ∞)

 image2  image1

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