Directed Reading for Content Mastery Section 1 Chemical Formulas and Equations Answers

Equations and Inequalities Involving Signed Numbers

In chapter 2 we established rules for solving equations using the numbers of arithmetic. Now that we have learned the operations on signed numbers, we will use those same rules to solve equations that involve negative numbers. Nosotros will besides study techniques for solving and graphing inequalities having one unknown.

SOLVING EQUATIONS INVOLVING SIGNED NUMBERS

OBJECTIVES

Upon completing this section y'all should be able to solve equations involving signed numbers.

Example 1 Solve for x and check: ten + 5 = 3

Solution

Using the same procedures learned in affiliate 2, we decrease 5 from each side of the equation obtaining

Example 2 Solve for x and cheque: - 3x = 12

Solution

Dividing each side by -3, we obtain

Ever cheque in the original equation.

Another fashion of solving the equation
3x - iv = 7x + eight
would exist to first decrease 3x from both sides obtaining
-4 = 4x + 8,
then decrease 8 from both sides and go
-12 = 4x.
At present divide both sides past 4 obtaining
- 3 = x or 10 = - 3.

Get-go remove parentheses. Then follow the process learned in affiliate two.

LITERAL EQUATIONS

OBJECTIVES

Upon completing this section you lot should exist able to:

1. Identify a literal equation.
2. Apply previously learned rules to solve literal equations.

An equation having more than than one letter of the alphabet is sometimes called a literal equation. It is occasionally necessary to solve such an equation for one of the messages in terms of the others. The step-by-step procedure discussed and used in chapter 2 is still valid afterwards any grouping symbols are removed.

Case ane Solve for c: iii(ten + c) - 4y = 2x - 5c

Solution

Beginning remove parentheses.

At this bespeak we notation that since we are solving for c, we want to obtain c on i side and all other terms on the other side of the equation. Thus we obtain

Call back, abx is the same every bit 1abx.
We divide by the coefficient of x, which in this case is ab.

Solve the equation 2x + 2y - 9x + 9a by get-go subtracting 2.v from both sides. Compare the solution with that obtained in the case.

Sometimes the grade of an answer tin can be changed. In this example we could multiply both numerator and denominator of the answer by (- l) (this does not modify the value of the answer) and obtain

The advantage of this last expression over the outset is that there are not then many negative signs in the respond.

Multiplying numerator and denominator of a fraction by the aforementioned number is a utilise of the fundamental principle of fractions.

The most commonly used literal expressions are formulas from geometry, physics, business organisation, electronics, and and then forth.

Instance four is the formula for the area of a trapezoid. Solve for c.

A trapezoid has two parallel sides and two nonparallel sides. The parallel sides are called bases.
Removing parentheses does not mean to simply erase them. We must multiply each term within the parentheses past the factor preceding the parentheses.
Changing the form of an respond is not necessary, merely you should be able to recognize when y'all accept a correct answer even though the form is not the same.

Example five is a formula giving interest (I) earned for a menstruation of D days when the principal (p) and the yearly charge per unit (r) are known. Find the yearly rate when the amount of interest, the chief, and the number of days are all known.

Solution

The problem requires solving for r.

Notice in this case that r was left on the right side and thus the computation was simpler. We can rewrite the answer another fashion if nosotros wish.

GRAPHING INEQUALITIES

OBJECTIVES

Upon completing this section y'all should be able to:

1. Apply the inequality symbol to represent the relative positions of two numbers on the number line.
2. Graph inequalities on the number line.

We have already discussed the set of rational numbers every bit those that can be expressed as a ratio of ii integers. At that place is likewise a set of numbers, called the irrational numbers,, that cannot be expressed every bit the ratio of integers. This set includes such numbers as so on. The set composed of rational and irrational numbers is called the real numbers.

Given any two real numbers a and b, information technology is always possible to state that Many times we are simply interested in whether or non two numbers are equal, but in that location are situations where nosotros as well wish to represent the relative size of numbers that are not equal.

The symbols < and > are inequality symbols or social club relations and are used to evidence the relative sizes of the values of ii numbers. We usually read the symbol < as "less than." For instance, a < b is read as "a is less than b." Nosotros usually read the symbol > equally "greater than." For instance, a > b is read equally "a is greater than b." Notice that we have stated that we usually read a < b as a is less than b. But this is only considering we read from left to right. In other words, "a is less than b" is the same as maxim "b is greater than a." Actually then, we take one symbol that is written two ways only for convenience of reading. One way to think the meaning of the symbol is that the pointed cease is toward the lesser of the two numbers.

The statement two < 5 tin exist read as "two is less than 5" or "five is greater than two."

a < b, "a is less than bif and only if there is a positive number c that can exist added to a to give a + c = b.

What positive number can exist added to 2 to give 5?

In simpler words this definition states that a is less than b if nosotros must add something to a to become b. Of course, the "something" must be positive.

If y'all think of the number line, you lot know that adding a positive number is equivalent to moving to the right on the number line. This gives rise to the following alternative definition, which may exist easier to visualize.

Example 1 three < half dozen, because three is to the left of half-dozen on the number line.

We could also write half-dozen > 3.

Example two - iv < 0, because -four is to the left of 0 on the number line.

Nosotros could also write 0 > - 4.

Case 3 four > - 2, considering 4 is to the correct of -two on the number line.

Example 4 - six < - 2, considering -6 is to the left of -two on the number line.

The mathematical statement x < 3, read as "x is less than 3," indicates that the variable x can be any number less than (or to the left of) 3. Remember, we are considering the existent numbers and not just integers, so exercise non call back of the values of x for x < 3 equally simply 2, 1,0, - 1, and then on.

Exercise you run into why finding the largest number less than 3 is impossible?

Every bit a matter of fact, to name the number x that is the largest number less than three is an impossible task. It can be indicated on the number line, nevertheless. To do this nosotros demand a symbol to represent the meaning of a statement such as 10 < 3.

The symbols ( and ) used on the number line indicate that the endpoint is not included in the ready.

Example 5 Graph x < 3 on the number line.

Solution

Note that the graph has an arrow indicating that the line continues without cease to the left.

This graph represents every real number less than 3.

Example 6 Graph x > iv on the number line.

Solution

This graph represents every real number greater than 4.

Example seven Graph 10 > -five on the number line.

Solution

This graph represents every real number greater than -5.

Example 8 Make a number line graph showing that 10 > - 1 and x < five. (The word "and" means that both conditions must utilise.)

Solution

The argument x > - 1 and 10 < five tin can be condensed to read - 1 < 10 < 5.

This graph represents all real numbers that are between - ane and five.

Example 9 Graph - three < x < 3.

Solution

If we wish to include the endpoint in the set, we apply a different symbol, :. We read these symbols as "equal to or less than" and "equal to or greater than."

Example ten x >; 4 indicates the number iv and all real numbers to the right of 4 on the number line.

What does ten < 4 represent?

The symbols [ and ] used on the number line bespeak that the endpoint is included in the prepare.

You will notice this utilise of parentheses and brackets to be consequent with their use in future courses in mathematics.

This graph represents the number one and all real numbers greater than 1.

This graph represents the number 1 and all existent numbers less than or equal to - 3.

Example thirteen Write an algebraic argument represented by the following graph.

Example 14 Write an algebraic statement for the following graph.

This graph represents all real numbers betwixt -four and 5 including -4 and 5.

Instance fifteen Write an algebraic statement for the following graph.

This graph includes 4 but not -two.

Example sixteen Graph on the number line.

Solution

This case presents a small trouble. How can nosotros indicate on the number line? If we estimate the point, and then another person might misread the statement. Could you possibly tell if the signal represents or possibly ? Since the purpose of a graph is to clarify, e'er label the endpoint.

A graph is used to communicate a argument. You should always name the nil bespeak to show direction and as well the endpoint or points to exist verbal.

SOLVING INEQUALITIES

OBJECTIVES

Upon completing this section you should be able to solve inequalities involving i unknown.

The solutions for inequalities generally involve the same basic rules as equations. There is 1 exception, which we volition soon observe. The first rule, however, is similar to that used in solving equations.

If the same quantity is added to each side of an inequality, the results are unequal in the aforementioned lodge.

Example 1 If 5 < 8, and so v + 2 < 8 + 2.

Example 2 If 7 < 10, then 7 - 3 < x - 3.

five + ii < 8 + two becomes 7 < 10.
vii - three < x - three becomes 4 < vii.

We can employ this rule to solve sure inequalities.

Instance three Solve for x: x + six < x

Solution

If we add -half-dozen to each side, we obtain

Graphing this solution on the number line, we have

Annotation that the procedure is the same equally in solving equations.

Nosotros will now use the addition rule to illustrate an important concept concerning multiplication or sectionalization of inequalities.

Suppose ten > a.

Now add - 10 to both sides by the addition rule.

Call back, adding the same quantity to both sides of an inequality does not alter its direction.

Now add together -a to both sides.

The last argument, - a > -ten, can be rewritten as - x < -a. Therefore we tin can say, "If x > a, and then - x < -a. This translates into the post-obit rule:

If an inequality is multiplied or divided by a negative number, the results will be diff in the opposite society.

For example: If 5 > 3 then -five < -iii.

Example 5 Solve for x and graph the solution: -2x>vi

Solution

To obtain x on the left side we must divide each term by - 2. Detect that since we are dividing by a negative number, we must modify the management of the inequality.

Notice that equally shortly as we carve up past a negative quantity, we must change the direction of the inequality.

Accept special notation of this fact. Each time you dissever or multiply by a negative number, you lot must modify the direction of the inequality symbol. This is the only difference between solving equations and solving inequalities.

When we multiply or divide past a positive number, at that place is no alter. When we multiply or divide by a negative number, the direction of the inequality changes. Exist careful-this is the source of many errors.

Once nosotros have removed parentheses and have just individual terms in an expression, the process for finding a solution is almost like that in chapter 2.

Let u.s.a. at present review the step-by-step method from chapter 2 and note the difference when solving inequalities.

Starting time Eliminate fractions by multiplying all terms by the least common denominator of all fractions. (No modify when we are multiplying by a positive number.)
2d Simplify by combining similar terms on each side of the inequality. (No change)
Third Add or subtract quantities to obtain the unknown on i side and the numbers on the other. (No change)
Fourth Divide each term of the inequality by the coefficient of the unknown. If the coefficient is positive, the inequality will remain the same. If the coefficient is negative, the inequality volition be reversed. (This is the important difference between equations and inequalities.)

The just possible difference is in the final step.

What must exist done when dividing past a negative number?

Don�t forget to label the endpoint.

SUMMARY

Cardinal Words

• A literal equation is an equation involving more than i letter of the alphabet.
• The symbols < and > are inequality symbols or society relations.
• a < b means that a is to the left of b on the real number line.
• The double symbols : betoken that the endpoints are included in the solution gear up.

Procedures

• To solve a literal equation for one letter in terms of the others follow the same steps equally in chapter two.
• To solve an inequality utilise the following steps:
  Step ane Eliminate fractions by multiplying all terms by the to the lowest degree mutual denominator of all fractions.
  Pace 2 Simplify by combining like terms on each side of the inequality.
  Footstep 3 Add or subtract quantities to obtain the unknown on i side and the numbers on the other.
  Step four Separate each term of the inequality by the coefficient of the unknown. If the coefficient is positive, the inequality volition remain the aforementioned. If the coefficient is negative, the inequality will be reversed.
  Step 5 Cheque your answer.
  

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