DrDelMath

Chapter 2: Equations, Inequalities, and Problem Solving

Section 2.1: Linear Equations in One Variable

Properties of Equations:

Linear Equations in One Variable:

Section 2.2: An Introduction to Problem Solving

Click HERE to see some worked exercises.
For some worked Mixture Problems click HERE.
And HERE are some more exercises and a few more HERE.

Section 2.3: Formulas and Problem Solving

Certain basic knowledge is required to do any mathematical modeling. For example to model a problem in mechanical engineering, you must know enough about mechanical engineering to understand the problem and convert it to mathematical statements. The very nature of the world in which we live requires that you must know and be able to use the basic formulas presented in this section.

Important Formulas

It is not required to memorize the following three formulas, but you should be able to work with them.

Percent

Each percent problem is solved by referring to a basic formula that relates percentage, percent, and base of the percentage.

Use this verbal statement and the basic percent formula to help translate percent problems into an equation. Finally remember the meaning of percent is "per 100" and convert all percents to decimals.

Click HERE to see some worked exercises.

Section 2.4: Linear Inequalities and Problem Solving

Properties of Inequalities:

Section 2.5: Compound Inequalities

Section 2.6: Absolute Value Equations

Solving Equations Involving Absolute Value

The basis for solving all equations involving absolute values is the definition of absolute value.

Notice the precise definition of absolute value has two cases:
    Case 1: The expression inside the absolute value symbol is positive or zero.
    Case 2: The expression inside the absolute value symbol is negative.

Procedure: To solve equations involving absolute values of variable expressions, it is necessary to solve the two equations which naturally result from the definition of absolute value. The solution set contains the solution set of the original equation. Therefore an integral part of the solution process is to test each of the possible solutions in the original equation.

Every equation involving absolute value is solved by considering the two cases as in this general example:
Generic Example: To solve an equation involving |something|, two cases must be considered. The two cases arise from the definition of absolute value. Therefore to solve any equation involving |something|, we consider:
    Case 1: The equation that results from relacing |something| with (something)
    Case 2: The equation that results from relacing |something| with (the opposite of something).

The following two properties of equations are also important tools when solving equations involving absolute value.

These two properties might be used to:
    Simplify the equation before considering the two cases.
    Solve the equations in Case 1 and Case 2.

In this section of the textbook the expresion inside the absolute value symbol is always a linear expresion in one variable. All of these equations can be solved using the two cases that come from the definition of absolute value together with the two properties for generating equivalent equations.

Solving an equation of the type |ax + b| = c.
     Consider the two cases.
     Case 1: (when ax + b is positive or zero) Solve the equation ax + b = c.
     Case 2: (when ax + b is negative) Solve the equation -(ax + b) = c.
In each case, the two properties for generating equivalent equations may be used to find a simplest equation.
The union of the solution sets for the two cases is the solution set for the original equation.

Solving an equation of the type |ax + b| + c = d.
     Add the expression -c to both sides to obtain the equivalent equation |ax + b| = d - c.
     Consider the two cases.
     Case 1: (when ax + b is positive or zero) Solve the equation ax + b = d - c.
     Case 2: (when ax + b is negative) Solve the equation -(ax + b) = d - c.
In each case, the two properties for generating equivalent equations may be used to find a simplest equation.
The union of the solution sets for the two cases is the solution set for the original equation.

Solving an equation of the type |ax + b| = |cx + d|.
     Consider the two cases.
     Case 1: (when ax + b is positive or zero) Solve the equation ax + b = cx + d.
     Case 2: (when ax + b is negative) Solve the equation -(ax + b) = cx + d.
In each case, the two properties for generating equivalent equations may be used to find a simplest equation.
The union of the solution sets for the two cases is the solution set for the original equation.

It may seem that there should be four cases in this last type of equation, but there are duplications as shown here.
     Case 1: (when ax + b is positive or zero) Solve the equation ax + b = |cx + d|.
         Case 1A: (when cx + d is positive or zero) Solve the equation ax + b = cx + d.
         Case 1B: (when cx + d is negative) Solve the equation ax + b = -(cx + d).
     Case 2: (when ax + b is negative) Solve the equation -(ax + b) = |cx + d|.
         Case 2A: (when cx + d is positive or zero) Solve the equation -(ax + b) = cx + d.
         Case 2B: (when cx + d is negative) Solve the equation -(ax + b) = -(cx + d).
If both sides of the equation in Case 2B is multiplied by -1, the equivalent equation obtained is identical to the equation in Case 1A.
If both sides of the equation in Case 2A is multiplied by -1, the equivalent equation obtained is identical to the equation in Case 1B.
For that reason only two cases are necessary.

Solving an equation of the type |ax + b| = c where c is a negative number.
Simply observe that absolute value is always non-negative (positive or zero) and therefore this type of equation does not have a solution. It solution set is the empty set.

Solving an equation of the type |ax + b| = 0.
The only way an absolute value can be zero is for the expression inside the absolute value symbol to be zero. So only Case 1 needs to be considered. No harm is done if Case 2 is considered, because in this type of equation it will always give the same solution set as Case 1.

Section 2.7: Absolute Value Inequalities