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Equations and Inequalities Involving Signed Numbers

In chapter 2 nosotros established rules for solving equations using the numbers of arithmetics. At present that we accept learned the operations on signed numbers, we will utilize those aforementioned rules to solve equations that involve negative numbers. Nosotros will also 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 10 and check: x + 5 = 3

Solution

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

Case 2 Solve for x and check: - 3x = 12

Solution

Dividing each side by -iii, nosotros obtain

Always check in the original equation.

Another way of solving the equation
3x - iv = 7x + 8
would be to offset subtract 3x from both sides obtaining
-4 = 4x + eight,
then subtract 8 from both sides and get
-12 = 4x.
At present carve up both sides past iv obtaining
- 3 = x or ten = - 3.

Beginning remove parentheses. Then follow the procedure learned in chapter 2.

LITERAL EQUATIONS

OBJECTIVES

Upon completing this section yous should be able to:

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

An equation having more than i 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 step-by-step procedure discussed and used in chapter 2 is still valid after any group symbols are removed.

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

Solution

First remove parentheses.

At this point we annotation that since nosotros are solving for c, we want to obtain c on 1 side and all other terms on the other side of the equation. Thus we obtain

Remember, abx is the same as 1abx.
We dissever past the coefficient of 10, which in this example is ab.

Solve the equation 2x + 2y - 9x + 9a by first subtracting 2.5 from both sides. Compare the solution with that obtained in the case.

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

The reward of this last expression over the starting time is that there are not then many negative signs in the answer.

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

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

Example 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 not hateful to only erase them. We must multiply each term inside the parentheses by the factor preceding the parentheses.
Changing the course of an respond is non necessary, but you should exist able to recognize when yous take a correct answer even though the class is not the same.

Example 5 is a formula giving interest (I) earned for a menstruum of D days when the primary (p) and the yearly charge per unit (r) are known. Find the yearly charge per unit when the amount of interest, the principal, and the number of days are all known.

Solution

The trouble requires solving for r.

Discover in this example that r was left on the correct side and thus the computation was simpler. We tin rewrite the answer another way if we wish.

GRAPHING INEQUALITIES

OBJECTIVES

Upon completing this section you lot 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.

We have already discussed the set up of rational numbers as those that can be expressed as a ratio of 2 integers. There is too a set of numbers, called the irrational numbers,, that cannot exist expressed every bit the ratio of integers. This set includes such numbers as and and then on. The gear up equanimous of rational and irrational numbers is called the real numbers.

Given any ii real numbers a and b, information technology is always possible to state that Many times we are only interested in whether or non two numbers are equal, but there 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 club relations and are used to show the relative sizes of the values of ii numbers. We normally read the symbol < as "less than." For example, a < b is read equally "a is less than b." We normally read the symbol > as "greater than." For case, a > b is read as "a is greater than b." Notice that we take stated that we usually read a < b as a is less than b. But this is merely 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, we have one symbol that is written two ways only for convenience of reading. One way to recollect the meaning of the symbol is that the pointed end is toward the bottom of the two numbers.

The argument 2 < v tin be read as "two is less than five" or "five is greater than two."

a < b, "a is less than bif and just if there is a positive number c that tin be added to a to requite a + c = b.

What positive number can 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 become b. Of course, the "something" must be positive.

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

Example 1 iii < half-dozen, because iii is to the left of 6 on the number line.

We could besides write half-dozen > 3.

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

We could also write 0 > - 4.

Case 3 4 > - 2, because 4 is to the correct of -ii on the number line.

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

The mathematical statement x < three, read as "ten is less than 3," indicates that the variable 10 tin can be any number less than (or to the left of) iii. Remember, we are considering the real numbers and not simply integers, and then practise non call up of the values of ten for ten < 3 as only 2, 1,0, - i, and then on.

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

As a matter of fact, to name the number x that is the largest number less than three is an impossible task. It can exist indicated on the number line, however. To do this we need a symbol to represent the significant of a statement such as x < iii.

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

Example five 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.

Instance 6 Graph 10 > iv on the number line.

Solution

This graph represents every real number greater than 4.

Example 7 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 10 > - 1 and x < 5. (The word "and" means that both weather must apply.)

Solution

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

This graph represents all existent numbers that are betwixt - 1 and v.

Case nine Graph - three < 10 < 3.

Solution

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

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

What does x < 4 represent?

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

You will find this use of parentheses and brackets to exist consistent with their use in future courses in mathematics.

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

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

Example 13 Write an algebraic statement represented past the post-obit graph.

Case 14 Write an algebraic statement for the post-obit graph.

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

Example xv Write an algebraic statement for the following graph.

This graph includes 4 but not -2.

Example 16 Graph on the number line.

Solution

This case presents a small problem. How can we indicate on the number line? If nosotros approximate the point, then another person might misread the statement. Could you possibly tell if the bespeak represents or maybe ? Since the purpose of a graph is to clarify, always label the endpoint.

A graph is used to communicate a argument. You lot should e'er proper noun the nada signal to show direction and likewise the endpoint or points to be 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 every bit equations. There is ane exception, which nosotros volition soon discover. The first rule, nonetheless, 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 order.

Case 1 If v < viii, then v + ii < 8 + ii.

Example ii If 7 < 10, then seven - iii < 10 - 3.

five + 2 < 8 + 2 becomes vii < ten.
vii - 3 < 10 - three becomes iv < 7.

We can use this rule to solve certain inequalities.

Example 3 Solve for 10: x + 6 < 10

Solution

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

Graphing this solution on the number line, we have

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

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

Suppose x > a.

Now 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.

Now add -a to both sides.

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

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

For example: If 5 > 3 then -v < -3.

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

Solution

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

Find that as soon equally we divide by a negative quantity, we must modify the management of the inequality.

Have special annotation of this fact. Each time you split or multiply past a negative number, you lot must change the direction of the inequality symbol. This is the only difference between solving equations and solving inequalities.

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

In one case we have removed parentheses and accept only individual terms in an expression, the process for finding a solution is well-nigh similar that in chapter ii.

Let u.s. at present review the step-by-step method from affiliate 2 and note the deviation when solving inequalities.

First Eliminate fractions past multiplying all terms by the least common denominator of all fractions. (No change when nosotros are multiplying by a positive number.)
Second Simplify by combining like terms on each side of the inequality. (No change)
Third Add together or decrease quantities to obtain the unknown on one side and the numbers on the other. (No change)
Fourth Split 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 exist reversed. (This is the important deviation between equations and inequalities.)

The merely possible divergence is in the terminal step.

What must be done when dividing by a negative number?

Don�t forget to characterization the endpoint.

SUMMARY

Key Words

• A literal equation is an equation involving more than one letter.
• The symbols < and > are inequality symbols or order relations.
• a < b means 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 gear up.

Procedures

• To solve a literal equation for one letter in terms of the others follow the same steps equally in affiliate 2.
• To solve an inequality utilise the following steps:
  Step 1 Eliminate fractions by multiplying all terms by the least common denominator of all fractions.
  Step 2 Simplify by combining similar terms on each side of the inequality.
  Step 3 Add or decrease quantities to obtain the unknown on 1 side and the numbers on the other.
  Step 4 Divide each term of the inequality past the coefficient of the unknown. If the coefficient is positive, the inequality will remain the same. If the coefficient is negative, the inequality will exist reversed.
  Step five Check your answer.
  

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