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CHAPTER 2
EQUATIONS, INEQUALITIES AND PROBLEM SOLVING
2.1First Degree Equations
An equality or equation is a statement where two symbols or groups of symbols are names for the same number.
A first-degree equation in one variable has an equals symbol, one variable and an (implied) exponent of one.
To solve a first-degree equation, we will follow two rules:
1)Whatever is done to one side of the equals must be done to the other.
2)Our goal is to group all the variables on one side of the equals and the constants on the other.
Word Problems (p. 55)
- Read the problem carefully and make sure you understand all terms.
- Read the problem a few more times to get an overview of the situation including what is given and what is being asked for.
- Sketch a figure, diagram or chart that might be helpful to describe the problem.
- Choose meaningful variables for each quantity in the problem. Try to represent all variables in terms of the unknown variable.
- Look for formulas or relationships in the statement of the problem.
- Write an equation.
- Solve the equation.
- Check your answers in the original statement of the problem.
2.2First Degree Equations with Fractions
Equations with fractions may be solved using the methods already learned in section 2.1 or by clearing all fractions from the problem as the first step.
To clear fractions: multiply every term by a common multiple of the denominators or, even better, by the LCM of the denominators.
2.3First Degree Equations with Decimals
Equations with decimals may be solved using the methods already learned in section 2.1 or by clearing all decimals from the problem as the first step.
To clear decimals: If you wrote each decimal as a fraction, you could multiply every term by a common multiple of the denominators or, even better, by the LCM of the denominators. There are two simpler ways to do this:
1)Multiply every term by a one followed by as many zeros as the maximum number of decimal places OR
2)Write every term with the same number of decimal places and then eliminate all decimals.
2.4Formulas
A formula typically has several variables and so it cannot give us an answer such as x = 5 since there are more variables. We might get an answer such as y = 2x – 7 or t = d/r that will make solving for that variable easier in the future.
2.5Inequalities
An inequality is a relationship much like an equality but one side of the “equation” is smaller than the other and one side is larger. Many values work as solutions but we are interested in finding the “endpoints” of these values.
For example, a U.S. citizen can vote at 18 years of age so we could say age 18 or a 18. It is obvious that a 50 year old can vote but we make the statement in terms of the endpoint.
All the rules of solving a first-degree equation work for a first-degree inequality EXCEPT
Multiplying OR dividing an inequality by a NEGATIVE number reverses the direction of the inequality!
There are three main ways to state an answer to an inequality:
1)Use a number line with circles to indicate the endpoints of the solution.
- For < or >, use an open circle.
- For or , use a closed circle.
2)Use a number line with brackets to indicate the endpoints of the solution.
- For < or >, use a rounded parenthesis “(“ or “)”. The direction will depend on the direction of the inequality.
- For or , use a square bracket [ or ].
3)Use interval notation. The low and high end of the possible range of numbers is listed between parentheses (for < or >) or brackets (for or ). Use when the range in one direction is infinite.
2.6Compound Statements
Conjunctions use the word and. All parts of a conjunction must be true for the entire statement to be true.
“I went to Baskin Robbins and I had a sundae” is only true if you had a sundae at Baskin Robbins.
If you had a sundae somewhere else or had a shake at Baskin Robbins or did not do either, the statement is false.
Disjunctions use the word “or”. Only one of the statements has to be true for the entire disjunction to be true.
“I went to Baskin Robbins or I had a hot dog” is only false if you did not have a hot dog and did not go to Baskin Robbins.
An intersection is indicated by the symbol . The intersection of Redondo Beach Blvd and Crenshaw Blvd is the area that is contained on both streets (i.e. the intersection). The intersection of x > 5 and x < 10 is the set of all numbers between 5 and 10 indicated by 5 < x < 10. The intersection of all people over 5’6” and 6’ tall is only the people over 6’ tall as they are both over 5’6” and over 6’.
The union is indicated by the symbol and contains all members of the sets including the intersection. Let’s say that the Lee Family has a son named Sean. Sean marries a woman named Liz Squid and they hyphenate their name to Squid-Lee. The union of the Lee Family and the Squid-Lee Family is Mr. and Mrs. Lee, Sean and Liz. The intersection is only Sean.
2.7Absolute Value
The absolute value of x is written as and can be viewed as the distance from x to zero on a number line. It can also be stated as the opposite of x ifx is negative. Since we do not know if the value within the absolute value is positive or negative, the problem is written as a disjunction.
- For an equality such as , the problem becomes x = k OR x = – k.
- For an inequality such as where k is a positive number, the problem becomes x < k OR x– k.
CHAPTER 3
GRAPHING LINEAR EQUATIONS AND INEQUALITIES
3.1The Rectangular Coordinate System
- Find an x or y and solve for the other variable. An easy choice is to use x = 0 and then solve for y and then to use y = 0 and then solve for x.
- Check for symmetry
y-axis symmetry: The graph is symmetric about the y-axis if replacing x with – xdoes not change the equation. The equations may need to be rearranged to match the original!
x-axis symmetry: The graph is symmetric about the x-axis if replacing y with – ydoes not change the equation.
Origin symmetry: The graph is symmetric about the origin if replacing x with – x AND y with – ydoes not change the equation.
- Find some additional ordered pairs that work in the original equation.
- Use any symmetry to reflect these pairs to their mirror image.
3.2Linear Equations in Two Variables