The Equation Y-100x=2000 Can Be Used to Represent

Equations and Inequalities Involving Signed Numbers

In affiliate 2 we established rules for solving equations using the numbers of arithmetics. Now that we take learned the operations on signed numbers, nosotros volition utilize those same rules to solve equations that involve negative numbers. We will likewise study techniques for solving and graphing inequalities having i unknown.

SOLVING EQUATIONS INVOLVING SIGNED NUMBERS

OBJECTIVES

Upon completing this section yous should exist able to solve equations involving signed numbers.

Example 1 Solve for x and check: x + five = iii

Solution

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

Example ii Solve for 10 and check: - 3x = 12

Solution

Dividing each side past -3, we obtain

E'er check in the original equation.

Another way of solving the equation
3x - 4 = 7x + 8
would exist to first subtract 3x from both sides obtaining
-iv = 4x + viii,
then decrease 8 from both sides and get
-12 = 4x.
Now divide both sides by 4 obtaining
- 3 = x or x = - 3.

First remove parentheses. So follow the procedure learned in chapter 2.

LITERAL EQUATIONS

OBJECTIVES

Upon completing this section you should be able to:

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

An equation having more than ane letter is sometimes called a literal equation. It is occasionally necessary to solve such an equation for ane of the messages in terms of the others. The step-by-pace procedure discussed and used in chapter 2 is all the same valid after whatsoever grouping symbols are removed.

Example 1 Solve for c: 3(x + c) - 4y = 2x - 5c

Solution

First remove parentheses.

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

Remember, abx is the same as 1abx.
We dissever by the coefficient of x, which in this case is ab.

Solve the equation 2x + 2y - 9x + 9a past outset subtracting 2.v from both sides. Compare the solution with that obtained in the example.

Sometimes the form of an reply can be changed. In this example we could multiply both numerator and denominator of the reply past (- l) (this does not change the value of the answer) and obtain

The advantage of this terminal expression over the first is that in that location are not so many negative signs in the reply.

Multiplying numerator and denominator of a fraction by the same number is a use of the central principle of fractions.

The nigh commonly used literal expressions are formulas from geometry, physics, business organization, electronics, and and so along.

Instance 4 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 chosen bases.
Removing parentheses does non hateful to just erase them. We must multiply each term inside the parentheses by the cistron preceding the parentheses.
Changing the grade of an reply is not necessary, but you should be able to recognize when you have a correct answer even though the grade is not the aforementioned.

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

Solution

The problem requires solving for r.

Notice in this example that r was left on the right side and thus the ciphering was simpler. We can rewrite the answer another way if nosotros wish.

GRAPHING INEQUALITIES

OBJECTIVES

Upon completing this section yous should be able to:

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

Nosotros have already discussed the set of rational numbers as those that tin exist expressed every bit a ratio of two integers. There is also a set of numbers, called the irrational numbers,, that cannot exist expressed as the ratio of integers. This set includes such numbers as and and then on. The set up composed of rational and irrational numbers is chosen the real numbers.

Given any two existent numbers a and b, it is always possible to land that Many times we are only interested in whether or not two numbers are equal, but there are situations where nosotros also wish to represent the relative size of numbers that are not equal.

The symbols < and > are inequality symbols or order relations and are used to show the relative sizes of the values of ii numbers. Nosotros usually read the symbol < every bit "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 every bit "a is greater than b." Detect that nosotros have stated that we unremarkably read a < b as a is less than b. Merely this is but because nosotros read from left to right. In other words, "a is less than b" is the same as saying "b is greater than a." Actually then, nosotros take 1 symbol that is written two ways only for convenience of reading. One manner to think the meaning of the symbol is that the pointed finish is toward the lesser of the two numbers.

The statement 2 < 5 can exist read as "ii is less than five" or "5 is greater than two."

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

What positive number tin can be added to ii to give 5?

In simpler words this definition states that a is less than b if we 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 calculation a positive number is equivalent to moving to the right on the number line. This gives rise to the post-obit alternative definition, which may exist easier to visualize.

Example ane iii < six, because 3 is to the left of half-dozen on the number line.

Nosotros could besides write 6 > 3.

Example 2 - 4 < 0, considering -4 is to the left of 0 on the number line.

We could too write 0 > - 4.

Example 3 4 > - 2, considering 4 is to the right of -2 on the number line.

Example 4 - 6 < - 2, because -6 is to the left of -2 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 non just integers, so do not think of the values of x for x < three every bit only 2, 1,0, - ane, and and then on.

Do you come across why finding the largest number less than 3 is incommunicable?

Every bit a affair of fact, to proper name the number x that is the largest number less than 3 is an incommunicable job. Information technology can be indicated on the number line, nonetheless. To do this we need a symbol to represent the meaning of a statement such as 10 < three.

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

Instance 5 Graph x < 3 on the number line.

Solution

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

This graph represents every real number less than 3.

Example 6 Graph x > 4 on the number line.

Solution

This graph represents every real number greater than 4.

Instance vii Graph 10 > -5 on the number line.

Solution

This graph represents every existent number greater than -5.

Example 8 Make a number line graph showing that x > - one and x < v. (The give-and-take "and" ways that both atmospheric condition must use.)

Solution

The statement x > - one and 10 < 5 can exist condensed to read - 1 < x < 5.

This graph represents all real numbers that are between - 1 and 5.

Instance nine Graph - 3 < x < 3.

Solution

If we wish to include the endpoint in the gear up, we use a dissimilar symbol, :. We read these symbols every bit "equal to or less than" and "equal to or greater than."

Example 10 x >; iv indicates the number 4 and all real numbers to the right of 4 on the number line.

What does x < iv represent?

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

You lot will find this apply of parentheses and brackets to be consistent with their apply in future courses in mathematics.

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

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

Example 13 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 between -4 and 5 including -4 and 5.

Instance 15 Write an algebraic statement for the following graph.

This graph includes 4 only not -ii.

Example 16 Graph on the number line.

Solution

This example presents a small trouble. How can we signal on the number line? If nosotros estimate the point, then another person might misread the statement. Could you possibly tell if the point represents or maybe ? Since the purpose of a graph is to analyze, always characterization the endpoint.

A graph is used to communicate a statement. You should always name the zero point to show direction and also the endpoint or points to exist exact.

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 one exception, which we volition presently observe. The first dominion, notwithstanding, 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 same lodge.

Example one If 5 < 8, and then 5 + 2 < 8 + 2.

Example ii If seven < x, then 7 - 3 < x - 3.

5 + 2 < 8 + 2 becomes 7 < x.
7 - 3 < 10 - 3 becomes 4 < 7.

We can use this rule to solve sure inequalities.

Example 3 Solve for x: x + 6 < 10

Solution

If we add -vi to each side, we obtain

Graphing this solution on the number line, nosotros take

Notation that the process is the same as in solving equations.

We will at present use the addition rule to illustrate an important concept concerning multiplication or segmentation of inequalities.

Suppose ten > a.

At present add together - ten to both sides past the addition rule.

Recall, adding the aforementioned quantity to both sides of an inequality does non change its direction.

At present add -a to both sides.

The last statement, - a > -x, can be rewritten as - x < -a. Therefore we can say, "If x > a, then - ten < -a. This translates into the following rule:

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

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

Case 5 Solve for ten and graph the solution: -2x>6

Solution

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

Notice that equally before long as we carve up by a negative quantity, we must change the management of the inequality.

Accept special note of this fact. Each time yous divide or multiply by a negative number, you must change the direction of the inequality symbol. This is the only difference betwixt solving equations and solving inequalities.

When we multiply or separate by a positive number, there is no change. When we multiply or separate by a negative number, the direction of the inequality changes. Be careful-this is the source of many errors.

In one case we accept removed parentheses and have only individual terms in an expression, the process for finding a solution is near like that in affiliate 2.

Permit united states of america at present review the footstep-by-pace method from chapter 2 and annotation the difference when solving inequalities.

First Eliminate fractions by multiplying all terms past the least common denominator of all fractions. (No modify when we are multiplying by a positive number.)
2d Simplify by combining like terms on each side of the inequality. (No change)
Third Add or decrease quantities to obtain the unknown on one 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 departure betwixt equations and inequalities.)

The just possible difference is in the final footstep.

What must be done when dividing by a negative number?

Don�t forget to label the endpoint.

SUMMARY

Key Words

• A literal equation is an equation involving more one letter.
• The symbols < and > are inequality symbols or order relations.
• a < b means that a is to the left of b on the existent number line.
• The double symbols : indicate that the endpoints are included in the solution prepare.

Procedures

• To solve a literal equation for 1 alphabetic character in terms of the others follow the same steps every bit in chapter 2.
• To solve an inequality use the post-obit steps:
  Step 1 Eliminate fractions by multiplying all terms by the to the lowest degree mutual denominator of all fractions.
  Step 2 Simplify by combining like terms on each side of the inequality.
  Footstep 3 Add together or decrease quantities to obtain the unknown on one side and the numbers on the other.
  Pace four Split up each term of the inequality past the coefficient of the unknown. If the coefficient is positive, the inequality will remain the aforementioned. If the coefficient is negative, the inequality will be reversed.
  Stride 5 Cheque your answer.
  

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