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Equations and Inequalities Involving Signed Numbers

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

SOLVING EQUATIONS INVOLVING SIGNED NUMBERS

OBJECTIVES

Upon completing this section you should be able to solve equations involving signed numbers.

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

Solution

Using the same procedures learned in chapter two, we subtract five from each side of the equation obtaining

Example 2 Solve for x 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 - four = 7x + 8
would be to showtime subtract 3x from both sides obtaining
-four = 4x + viii,
then subtract 8 from both sides and get
-12 = 4x.
At present split both sides by 4 obtaining
- 3 = 10 or x = - iii.

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

LITERAL EQUATIONS

OBJECTIVES

Upon completing this department you should be able to:

1. Place a literal equation.
2. Use previously learned rules to solve literal equations.

An equation having more than one letter is sometimes called a literal equation. Information technology is occasionally necessary to solve such an equation for one of the letters in terms of the others. The pace-by-pace procedure discussed and used in chapter 2 is nevertheless valid after any group symbols are removed.

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

Solution

First remove parentheses.

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

Recollect, abx is the same as 1abx.
We carve up past the coefficient of x, which in this case is ab.

Solve the equation 2x + 2y - 9x + 9a by first subtracting ii.v from both sides. Compare the solution with that obtained in the instance.

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

The advantage of this last expression over the commencement is that at that place are not then many negative signs in the answer.

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

The most usually used literal expressions are formulas from geometry, physics, business, electronics, and and so along.

Example 4 is the formula for the area of a trapezoid. Solve for c.

A trapezoid has ii parallel sides and two nonparallel sides. The parallel sides are chosen bases.
Removing parentheses does not mean to simply erase them. We must multiply each term inside the parentheses past the gene preceding the parentheses.
Irresolute the form of an respond is not necessary, but yous should exist able to recognize when y'all have a right answer even though the form is non the same.

Example 5 is a formula giving interest (I) earned for a period of D days when the chief (p) and the yearly rate (r) are known. Find the yearly rate when the amount of interest, the main, 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 correct side and thus the ciphering was simpler. We can rewrite the respond another manner if we wish.

GRAPHING INEQUALITIES

OBJECTIVES

Upon completing this section you lot should be able to:

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

We take already discussed the prepare of rational numbers as those that tin can be expressed as a ratio of two integers. There is besides a fix of numbers, called the irrational numbers,, that cannot be expressed equally the ratio of integers. This ready includes such numbers equally so on. The set equanimous of rational and irrational numbers is called the existent numbers.

Given whatsoever two real numbers a and b, it is always possible to state that Many times we are only interested in whether or not two numbers are equal, but at that place are situations where we likewise wish to represent the relative size of numbers that are not equal.

The symbols < and > are inequality symbols or society relations and are used to show the relative sizes of the values of 2 numbers. We usually read the symbol < as "less than." For case, a < b is read as "a is less than b." We normally read the symbol > every bit "greater than." For instance, a > b is read as "a is greater than b." Notice that we have stated that nosotros ordinarily read a < b as a is less than b. But this is simply because we read from left to right. In other words, "a is less than b" is the same every bit maxim "b is greater than a." Actually then, nosotros accept 1 symbol that is written two means simply for convenience of reading. One way to remember the pregnant of the symbol is that the pointed finish is toward the lesser of the two numbers.

The statement 2 < 5 tin be read as "two is less than v" or "five 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 be 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 go b. Of course, the "something" must be positive.

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

Case i three < half dozen, because 3 is to the left of 6 on the number line.

We could also write 6 > 3.

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

We could also write 0 > - four.

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

Example 4 - half dozen < - 2, because -6 is to the left of -2 on the number line.

The mathematical statement x < 3, read every bit "x is less than three," indicates that the variable x can be whatever number less than (or to the left of) 3. Retrieve, nosotros are because the existent numbers and non only integers, so practice not think of the values of 10 for ten < 3 as only 2, i,0, - 1, and so on.

Do you come across why finding the largest number less than three is impossible?

Every bit a matter of fact, to name the number x that is the largest number less than 3 is an impossible task. It can exist indicated on the number line, withal. To practise this we need a symbol to correspond the meaning of a argument such equally x < iii.

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

Example v Graph x < 3 on the number line.

Solution

Annotation that the graph has an arrow indicating that the line continues without finish to the left.

This graph represents every real number less than 3.

Example vi Graph x > iv on the number line.

Solution

This graph represents every real number greater than 4.

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

Solution

This graph represents every real number greater than -5.

Example viii Make a number line graph showing that ten > - i and ten < v. (The discussion "and" means that both conditions must utilize.)

Solution

The statement ten > - i and x < 5 can be condensed to read - one < 10 < v.

This graph represents all existent numbers that are betwixt - i and 5.

Example 9 Graph - 3 < x < three.

Solution

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

Case 10 ten >; four indicates the number 4 and all real numbers to the correct of 4 on the number line.

What does 10 < 4 stand for?

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

You will discover this utilise of parentheses and brackets to be consistent with their utilise in time to come courses in mathematics.

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

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

Example xiii Write an algebraic statement represented by the following graph.

Example xiv Write an algebraic statement for the following graph.

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

Example fifteen Write an algebraic statement for the following graph.

This graph includes 4 but non -two.

Example 16 Graph on the number line.

Solution

This instance presents a pocket-sized problem. How tin we indicate on the number line? If we approximate the point, then some other person might misread the argument. Could you lot possibly tell if the indicate represents or maybe ? Since the purpose of a graph is to clarify, always characterization the endpoint.

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

SOLVING INEQUALITIES

OBJECTIVES

Upon completing this department you should be able to solve inequalities involving one unknown.

The solutions for inequalities more often than not involve the same basic rules equally equations. At that place is one exception, which we will soon discover. The kickoff dominion, even so, is similar to that used in solving equations.

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

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

Case 2 If 7 < 10, then seven - three < ten - iii.

5 + two < 8 + 2 becomes 7 < 10.
7 - 3 < 10 - 3 becomes iv < 7.

We can use this dominion to solve certain inequalities.

Instance 3 Solve for 10: 10 + vi < ten

Solution

If we add -6 to each side, we obtain

Graphing this solution on the number line, we have

Note that the process is the aforementioned as in solving equations.

We volition now use the addition rule to illustrate an important concept concerning multiplication or division of inequalities.

Suppose 10 > a.

At present add - x to both sides past the add-on rule.

Remember, adding the same quantity to both sides of an inequality does not change its direction.

At present add together -a to both sides.

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

If an inequality is multiplied or divided past a negative number, the results will be unequal in the opposite order.

For example: If 5 > iii so -5 < -3.

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

Solution

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

Observe that as presently equally we divide by a negative quantity, we must change the direction of the inequality.

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

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

Once we take removed parentheses and have but individual terms in an expression, the procedure for finding a solution is nearly like that in chapter two.

Let us now review the step-past-step method from chapter two and annotation the difference when solving inequalities.

Offset Eliminate fractions by multiplying all terms by the least common denominator of all fractions. (No modify when we are multiplying by a positive number.)
Second Simplify by combining like terms on each side of the inequality. (No modify)
Third Add together or subtract quantities to obtain the unknown on 1 side and the numbers on the other. (No change)
Fourth Carve up each term of the inequality by 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. (This is the of import difference between equations and inequalities.)

The merely possible difference is in the last step.

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 than than ane alphabetic character.
• The symbols < and > are inequality symbols or order relations.
• a < b ways that a is to the left of b on the real number line.
• The double symbols : indicate that the endpoints are included in the solution prepare.

Procedures

• To solve a literal equation for 1 letter in terms of the others follow the same steps as in chapter 2.
• To solve an inequality employ the post-obit steps:
  Step ane Eliminate fractions by multiplying all terms by the to the lowest degree mutual denominator of all fractions.
  Footstep 2 Simplify by combining like terms on each side of the inequality.
  Pace 3 Add or decrease quantities to obtain the unknown on one side and the numbers on the other.
  Step iv Split 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 exist reversed.
  Footstep five Bank check your answer.
  

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