What is the solution to the inequality 3t 9 ≥ 15 ?t ≥

Equations and Inequalities Involving Signed Numbers

In affiliate 2 we established rules for solving equations using the numbers of arithmetic. Now that we take learned the operations on signed numbers, we will use those aforementioned rules to solve equations that involve negative numbers. We will too study techniques for solving and graphing inequalities having ane unknown.

SOLVING EQUATIONS INVOLVING SIGNED NUMBERS

OBJECTIVES

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

Example 1 Solve for ten and check: x + v = three

Solution

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

Example ii Solve for ten and cheque: - 3x = 12

Solution

Dividing each side past -3, we obtain

Always bank check in the original equation.

Another mode of solving the equation
3x - four = 7x + 8
would be to first subtract 3x from both sides obtaining
-4 = 4x + eight,
and then subtract eight from both sides and become
-12 = 4x.
Now dissever both sides by 4 obtaining
- iii = x or x = - three.

First remove parentheses. Then follow the procedure learned in affiliate 2.

LITERAL EQUATIONS

OBJECTIVES

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

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

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

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

Solution

First remove parentheses.

At this point 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 divide by the coefficient of x, which in this case is ab.

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

Sometimes the class of an answer tin 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 there are not and then many negative signs in the respond.

Multiplying numerator and denominator of a fraction past the aforementioned number is a use of the fundamental principle of fractions.

The nigh unremarkably used literal expressions are formulas from geometry, physics, business, electronics, and then along.

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

A trapezoid has two parallel sides and ii nonparallel sides. The parallel sides are called bases.
Removing parentheses does not mean to merely erase them. We must multiply each term inside the parentheses past the factor preceding the parentheses.
Irresolute the class of an answer is not necessary, simply you should be able to recognize when you take a correct reply fifty-fifty though the form is not the same.

Example five is a formula giving involvement (I) earned for a flow of D days when the principal (p) and the yearly rate (r) are known. Find the yearly rate when the corporeality of interest, the main, and the number of days are all known.

Solution

The problem requires solving for r.

Discover in this case that r was left on the right side and thus the ciphering was simpler. We tin rewrite the answer another fashion if we wish.

GRAPHING INEQUALITIES

OBJECTIVES

Upon completing this section you 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 of rational numbers equally those that tin can be expressed equally a ratio of two integers. There is also a set of numbers, chosen the irrational numbers,, that cannot be expressed as the ratio of integers. This set includes such numbers as and and so on. The set composed of rational and irrational numbers is called the real numbers.

Given any 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 as well wish to correspond the relative size of numbers that are not equal.

The symbols < and > are inequality symbols or society relations and are used to bear witness the relative sizes of the values of 2 numbers. We usually read the symbol < as "less than." For instance, a < b is read as "a is less than b." We commonly read the symbol > as "greater than." For example, a > b is read as "a is greater than b." Notice that we accept stated that nosotros ordinarily read a < b as a is less than b. But this is only because nosotros read from left to right. In other words, "a is less than b" is the aforementioned every bit proverb "b is greater than a." Actually then, nosotros have one symbol that is written two ways only for convenience of reading. One way to remember the meaning of the symbol is that the pointed stop is toward the lesser of the 2 numbers.

The argument 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 simply if there is a positive number c that tin can be added to a to give a + c = b.

What positive number can be added to ii 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 lot know that calculation a positive number is equivalent to moving to the right on the number line. This gives rise to the following alternative definition, which may be easier to visualize.

Example 1 3 < 6, considering 3 is to the left of 6 on the number line.

Nosotros could also write 6 > 3.

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

We could also write 0 > - 4.

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

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

The mathematical statement x < 3, read as "x is less than iii," indicates that the variable x can be any number less than (or to the left of) three. Remember, we are because the real numbers and non just integers, so do not recall of the values of x for 10 < iii as only two, one,0, - 1, and then on.

Practice you see why finding the largest number less than iii is impossible?

As a matter of fact, to name the number x that is the largest number less than iii is an impossible chore. Information technology can exist indicated on the number line, however. To do this we demand a symbol to represent the meaning of a argument such as 10 < 3.

The symbols ( and ) used on the number line indicate that the endpoint is non included in the set.

Case 5 Graph x < three on the number line.

Solution

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

This graph represents every real number less than 3.

Example 6 Graph x > four on the number line.

Solution

This graph represents every existent number greater than 4.

Case vii Graph 10 > -5 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 < 5. (The word "and" means that both conditions must use.)

Solution

The statement x > - 1 and ten < 5 tin can be condensed to read - 1 < ten < 5.

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

Example nine Graph - three < x < 3.

Solution

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

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

What does 10 < iv represent?

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

Y'all will detect this apply of parentheses and brackets to be consequent with their use in future courses in mathematics.

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

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 post-obit graph.

Example 14 Write an algebraic argument for the following graph.

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

Example 15 Write an algebraic statement for the following graph.

This graph includes 4 simply not -2.

Example 16 Graph on the number line.

Solution

This example presents a small-scale problem. How can we indicate on the number line? If we gauge the bespeak, then another person might misread the statement. Could you lot possibly tell if the signal represents or peradventure ? Since the purpose of a graph is to clarify, e'er label the endpoint.

A graph is used to communicate a statement. You lot should ever proper noun the naught point to show direction and also the endpoint or points to exist verbal.

SOLVING INEQUALITIES

OBJECTIVES

Upon completing this department you should be able to solve inequalities involving 1 unknown.

The solutions for inequalities generally involve the same basic rules as equations. At that place is one exception, which we will soon discover. The showtime 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 diff in the aforementioned order.

Instance 1 If 5 < 8, then 5 + 2 < viii + ii.

Case 2 If vii < 10, then 7 - 3 < 10 - 3.

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

We tin can apply this rule to solve certain inequalities.

Example 3 Solve for ten: x + 6 < 10

Solution

If we add -6 to each side, we obtain

Graphing this solution on the number line, we have

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

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

Suppose x > a.

Now add - x to both sides past the addition rule.

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

Now add -a to both sides.

The final argument, - a > -x, tin be rewritten equally - x < -a. Therefore we can say, "If x > a, then - x < -a. This translates into the following rule:

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

For instance: If v > three and then -five < -3.

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

Solution

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

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

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

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

Once we accept removed parentheses and accept just individual terms in an expression, the process for finding a solution is nigh like that in chapter two.

Allow the states now review the footstep-by-step method from chapter ii and note the difference when solving inequalities.

First Eliminate fractions past multiplying all terms by the least mutual denominator of all fractions. (No change when we are multiplying by a positive number.)
2nd Simplify by combining like terms on each side of the inequality. (No change)
Third Add together or subtract quantities to obtain the unknown on one side and the numbers on the other. (No alter)
4th Split 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 important difference betwixt equations and inequalities.)

The only possible difference is in the final step.

What must be washed 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 of the alphabet.
• The symbols < and > are inequality symbols or guild relations.
• a < b means that a is to the left of b on the existent number line.
• The double symbols : signal that the endpoints are included in the solution set.

Procedures

• To solve a literal equation for 1 letter of the alphabet in terms of the others follow the same steps equally in chapter ii.
• To solve an inequality use the following steps:
  Pace 1 Eliminate fractions by multiplying all terms past the least common denominator of all fractions.
  Footstep 2 Simplify past 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 4 Split 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.
  Stride 5 Check your reply.
  

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