The first step in solving real-world problems mathematically is to change the problem into an equation. Sometimes, it is a matter of deciding what operations need to be used. It is also a matter of seeing how quantities relate to one another.

Words and phrases that represent addition include such things as added to, increased from, more than, or the sum of. For example, 9 added to a number could be represented algebraically as 9 + x. Words and phrases that represent subtraction include, subtracted from, less than, decreased by, or the difference between. Statements such as multiplied by, the product of, twice or three times, or a certain percent indicate multiplication, and statements such as divided by or the quotient of can indicate division. Sometimes a statement can indicate more than one operation. For example, a statement such as 5 less than 3 times a number could be represented as 3x – 5.

Figure 1: Some words that suggest mathematical operations.

It is also important to determine the relationship between quantities. Suppose that the length of a rectangle is 5 inches longer than its width. The width of the rectangle could be represented as w, so the length is w + 5. When one quantity is given in terms of another, it is usually easier to let one variable stand for the base amount. Any letter other than x can be used, as long as what the variable stands for is clearly defined.

Suppose that one notebook costs $3.00. Then 5 notebooks would cost 5 ∙ 3 or $15.00. If one textbook costs x, then 5 copies of the same text would cost 5x. Similarly, the cost of x items at y dollars can be represented as xy. Sometimes a statement can include more than one operation. Suppose the cost to rent a truck included a daily fee of $40.00, as well as a mileage fee of 60 cents a mile. If the truck were being rented for 2 days, the total cost of the truck would be the daily fee of 40.00 ∙ 2 for the number of days, plus the mileage fee of 0.60x.

Figure 2: Using multiplication and addition to solve a problem.

The word is often means is equal to. Suppose in the truck rental example, the total cost of the truck rental is $260.00. Then $80.00 + 0.60x = 260.00. The equation could then be solved as .60x = 260 – 80, or .60x = 180, or x = 180/.60 = 300 miles.

Figure 3: An example of translating an application to an equation.

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]]>Real numbers include rational and irrational numbers, whole numbers, positive and negative integers, and the number 0. They can be used in operations such as addition, subtraction, multiplication, and division, as well as in expressions involving variables.

Real numbers include many different types of numbers that are familiar, such as positive and negative integers, along the number line. Rational numbers, such as 3/5, are also included, as well as irrational numbers such as π, √7, and √2. The representation of infinity ∞ is not a real number, nor are imaginary numbers.

Figure 1: Contents of the set of real numbers.

Since real numbers fall on the number line, they can be added using the number line model. For example, to add 3 + 2, go from 0 to 3 on the number line, and then forward 2 spaces to 5. Similarly, adding -2 + -5, go left on the number line 2 spaces to -2 and then go left 5 more, to -7. Fractions can be added by finding the least common denominator, then adding them using the number line, because fractions are rational numbers.

Figure 2: Addition of real numbers using the number line model.

Subtraction is the same thing as adding the additive inverse of the number to be subtracted. In symbol form, for any two numbers a and b, a – b is the same thing as a + (-b). Recall that the additive inverse of a number is the opposite of that number, so that a + (-a) = 0, and b + (-b) = 0. Subtraction can also be demonstrated using the real number line model, so that subtracting 7-6 is a matter of starting at 0 and going 7 spaces to 7, then going to the left 6 spaces, to 1.

Figure 3: Definition of subtraction in symbol form.

There are two important rules to remember when multiplying numbers. The product of two positive numbers or two negative numbers is a positive number. The product of a positive number and a negative number is a negative number. For example, 6∙ 2 is 12, as the product of two positive numbers. Similarly -5 ∙ -6 is 30, as the product of two negative numbers. However, -7 ∙ 3 is -21, as the product of a negative number and a positive number, and 4 ∙ -7 is -28, as the product of a positive number and a negative number. The rules for dividing numbers are similar. If both numbers are positive, the quotient will be positive, or if both numbers are negative, the quotient is positive.

Figure 4: Rules for multiplication and division of real numbers.

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]]>Decimals and percent are used in many different types of mathematical calculations and applications. They require attention to detail to use correctly.

In order to add and subtract numbers that contain decimal points, the numbers must be aligned so that the place values are the same. Then they can be added or subtracted as if they were whole numbers. Suppose 3.4 + 90.030 + 5.14 were to be added. In order to line up the numbers around the decimal point, 3.4 could also be written as 3.400, and 5.14 could be written as 5.140. Then 3.400 + 90.030 + 5.140 are all lined up, and equal 98.570. Subtraction is similar. 34.07 -1.569 could be rewritten as 34.070 to line up the decimal points, and then subtract 1.569 to equal 32.501.

Figure 1: Line up decimal points for addition and subtraction.

When multiplying numbers containing decimal points, factors can be multiplied as if they were whole numbers. Then, the number of decimal places in the answer is determined by the number of decimal places to the right in both factors. Suppose that 2.99 is multiplied by 4.8. The answer, 14.352, has three decimal places to the right, two from the factor 2.99 and one from the factor 4.8. Division is the reverse process, with an additional step. Suppose that 12.6 is divided by 2.24. In order to make 2.24 a whole number, both the numerator and the denominator must be multiplied by 100. Therefore, the new ratio is (12.6 ∙ 100)/(2.24 ∙ 100) = 1260/224 = 5.63. Multiplying the numerator and the denominator by 100/100 is the same thing as multiplying by 1, the identity element for multiplication.

When rounding decimals, the place value is always specified. Suppose 36.266 is to be rounded to the nearest hundredth. If the digit to the right of the hundredth place, the thousandth place, is more than 5, then the number in the hundredth place will be rounded up to the next number, or 36.27. Suppose the number 36.313 is to be rounded to the nearest hundredth. The digit 3 in the thousandth place is less than 5. The number remains as 36.31, and the digit in the thousandth place drops off, or the number is truncated.

The word **percent** means “per hundred”. Suppose that the decimal is 0.7. For the percent, the decimal is moved two spaces to the right, or multiplied by 100. If necessary, zeroes are added, so that .7 is 70%. Similarly, .35 is 35%, and 0.012 is 1.2 %.

Figure 3: For decimal to percent, move the decimal two spaces to the right.

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]]>Cube and fourth roots follow rules similar to those for square roots. The rules for rational exponents are an extension of the rules for exponents and roots.

If any real number b^{3} equals c, then b is the cube root of c, when b is not equal to 0. For example, 2^{3} equals 8, so 2 is the cube root of 8. Similarly, 3^{3} equals 27, so 3 is the cube root of 27. The cube root of a positive number is a positive number, because a positive times a positive times a positive is a positive. A negative number, such as -125, can have a real cube root, such as -5, because -5^{3} is -125. Although a negative number times a negative number is a positive number, that positive number times a negative number is a negative number.

Figure 1: The definition of cube roots in symbol form.

If any real number b^{4} equals c, then b is the fourth root of c, when c is a positive number. For example, 4^{4} is equal to 256, so 4 is the fourth root of 256. Similarly, 10^{4} equals 10,000, so 10 is the fourth root of 10,000. The fourth root of a positive number is a positive number. The fourth root of a negative number is not a real number, similar to the square root of a negative number.

A rational exponent such as x^{1/2} is the same thing as the square root of x, √x. Similarly, a rational exponent such as x^{1/3} is the same thing as the cube root of x, and x^{1/4}, as the fourth root of x. The definition can be extended to other rational exponents as well. For example, 8^{2/3} means the same thing as the cube root of 8, which is 2, squared which is 4. Recall that the cube root of 8 is 2, and that 2^{1} is equal to 2. Therefore, x^{1/3} is the cube root of x^{1}, and x^{1} equals x for all real numbers.

Figure 2: Rational exponents of real numbers.

Recall that a negative exponent, a^{-x}, means the same thing as 1/(a^{x}), so that 3^{-2} means the same thing as 1/3^{2}, or 1/9, and so on. That definition holds true for negative rational exponents, so that 27^{-2/3}, means the same thing as 1 over the cube root of 27, which is 3, squared. Then, (1/3)^{2} is 1/9.

Figure 3: Solving a negative rational exponent.

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]]>Linear inequalities in one variable can be solved similarly to linear equations in one variable. Properties that hold for linear inequalities include properties of addition, subtraction, multiplication, and division, as well as those involving relationships between negative numbers.

There are four properties of inequalities that hold true for real numbers. The addition property of equality in symbol form is: if a > b then a + c > b + c. Similarly, for subtraction, if a > b, then a – c > b –c. For the properties of multiplication and division, c must be greater than 0. If a > b and c > 0, then ac > bc. Similarly, if a >b and c > 0, then a/c > b /c. Suppose that the inequality is x -5 >-2. The variable can be isolated on the left side of the equation to solve it, so that x > -2 + 5, or x > 3. Also, a > b means the same thing as b < a.

Figure 1: Properties of inequality in symbol form.

The properties of inequalities for multiplication and division specify that the real number c is greater than 0 for the relationship to hold. What happens if c is a negative number? Suppose the inequality is 5 > 3, and both sides were multiplied by -4. 5(-4) is -20 and 3(-4) is -12. The relationship is -20 < -12, so the direction of the sign changes. Similarly, suppose that 100 > 80, and both sides were divided by -10. 100/-10 is -10 and 80/-10 = -8. The relationship is -10 < -8, so the direction of the sign also changes for division.

Inequalities with the variable on both sides of the sign are solved similarly to equations with the variable on both sides of the equals sign. Suppose the expression is -5q + 9 < -2q + 6. To solve the inequality, -5q + 2q + 9 < -2q + 2q + 6, or -3q + 9 < +6, or -3q < 6 – 9, or -3q < -3. Since dividing by -3 will change the sign, q > 1.

Figure 2: Process of solving inequalities when the variable is on both sides of the sign.

Similar to equations with all real numbers as the solution set, some inequalities have all real numbers as their solution set. Suppose the inequality is 2(x + 3) ≤ 5x -3x + 8. It can be simplified as 2x + 6 ≤ 2x + 8, or 6 ≤ 8. The statement is true for all real numbers.. Suppose the inequality is 3(x + 1) > 3x + 5. Then 3x + 3 > 3x + 5. There is no real number such that 3 > 5, so there is no solution.

Figure 3: An example of an inequality that is true for all real numbers.

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]]>Linear equations can be solved by isolating the variable on one side of the equals sign. This can be done by using the addition and multiplication properties, whether the variable is on one side of the equals sign or there is a variable on both sides of the equals sign.

The goal when solving an equation is to isolate the variable on one side of the equation. That way, the constants are on the other side of the equals sign and the variable is on the other. The variable can be isolated by using the properties of addition, multiplication, or the distributive property of real numbers. Suppose the equation to be solved is 2x + 4 = 20. The variable can be isolated by using the property of addition on both sides of the equation, as 2x + 4 – 4 = 20 – 4. Then 2x = 16. Both sides of the equation can then be divided by 2 so (2x)/2 = 16/2, or x = 8. The solution can be checked by the original equation: 2∙8 + 4 = 20, which is a true statement.

Figure 1: Steps for solving equations.

Like terms or parentheses should be eliminated before isolating the variable on one side of the equation. Suppose the equation were 42x + 3 + 2x + 5 – 8 = 88. The first step would be to combine the like terms using the commutative property, so that 42x + 2x + 3 + 5 – 8 = 88, or 44x + 0 = 88. The variable is easy to isolate as 0 is the identity for addition and can be eliminated, so that 44x = 88. Dividing both sides by 2, using the multiplication property, (44x)/44 = 88/44, or x = 2. Checking the solution, 84 + 3 + 4 + 5 -8 = 88. Suppose the equation were 3y – (2y + 5) = 7. The parentheses can be eliminated first as 3y -2y -5 = 7. Combining like terms, y – 5 = 7 or y = 12.

Solving linear equations when the variable on both sides of the equals sign is similar to solving a linear equation when the variable is on one side of the equals sign. The goal is still to isolate the variable on one side of the equation. Suppose the equation is 5x + 11 = 2x + 20. The variables can be isolated on one side or the other. If the variables are to be located on the left side, 5x – 2x + 11 = 2x – 2x + 20. Combining the like terms in variables, 3x + 11 = 20. Then, 3x = 20 – 11, or 3x = 9. Dividing both sides by 3, (3x)/3 = 9/3, or x = 3. If the variable is to be isolated on the right side, 5x -5x + 11 = 2x -5x + 20. Then, 11 = -3x + 20, 11 – 20 = -3x or -9 =-3x, (-3x)/3 = -9. Since a negative times a positive is a negative, x = 3.

Figure 2: Solving equations with the variable on both sides of the equals sign.

Some linear equations with the variables on both sides have more than one solution. Suppose the equation to be solved is 6x – 3 = 2(3x – 1). Eliminating the parentheses, 6x -3 = 6x -3. Since the equations on both sides of the equals sign are the same equation, any solution for x will be a correct solution. If x = 0, -3 = -3. If x = 1, then 3 = 3. If x = 2, then 9 = 9, and so on. The solution for x is all real numbers. In contrast, some linear equations with the variable on both sides have no solutions. Suppose the equation were 4x + 9 = 2(2x + 5). Eliminating the parentheses, 4x + 9 = 4x + 10. Isolating the variable on one side, 4x -4x + 9 = 4x – 4x + 10 or 9 = 10. That is a nonsense statement, and means that the problem has no solution.

Figure 3: An equation with no solution.

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Real numbers are numbers on the number line. This includes positive and negative numbers, integers, rational numbers, irrational numbers, and the number zero. Real numbers have certain mathematical properties, including commutative, associative, distributive, identity, and inverse properties.

The commutative properties hold for addition and multiplication of real numbers. In symbol form, the Commutative Property for Addition is stated as a + b = b + a. For example, 2 + 4 = 4 + 2. Also, 3 + -4 = -4 + 3. The Commutative Property for Multiplication in symbol form is a∙b = b∙a. For example, 5 ∙ 7 is the same thing as 7 ∙ 5. The commutative property does not hold for subtraction or division. In symbol form, a-b is not the same thing as b-a. For example, 5-3 = 2, but 3-5 = -2, not the same. Also, a/b is not the same thing as b/a. For example, 5/20 equals 1/4, but 20/5 equals 4.

Figure 1: The Commutative Properties for Addition and Multiplication.

The Associative Property of Addition states that for any real numbers a, b, and c, (a + b) + c = a + (b + c). For example, (5 + 6) + 7 equals 11 + 7 or 18. 5 + (6 +7) equals 5 + 13, which also equals 18. The groups of numbers have just changed. The Associative Property of Multiplication states that for any real numbers a, b, and c, (a ∙ b) ∙ c = a ∙ (b ∙ c). For example, (3∙ 2) ∙ 5 equals 6∙ 5, or 30, and 3 ∙ (2∙5) equals 3∙ 10, or 30.

Figure 2: The Associative Properties for Addition and Multiplication.

The Distributive Property involves 2 operations, multiplication and addition. In symbol form, a (b + c) = a∙b + a∙ c. For example, 5(2 + 2) = 20, and so does 10 + 10. There is also a Distributive Property of Multiplication over Subtraction such that a ∙ (b-c) = a∙ b –a ∙c. For example, 7 ∙ (5 -2) = 21, and so does 35 – 14.

Figure 3: The Distributive Property distributes multiplication over addition (or subtraction).

The identity element for addition is 0. Any real number plus 0 equals the number. The identity element for multiplication is 1. Any real number times 1 equals the number. The additive inverse for any real number a is –a, and a + -a = 0, the identity element for addition. The multiplicative inverse for a, a not equal to 0, is its reciprocal, 1/a. In symbol form, a ∙ 1/a = 1, the identity element for multiplication.

Figure 4: Adding a real number and its additive inverse = 0, the additive identity, and multiplying a real number times its multiplicative inverse = 1, the multiplicative identity.

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]]>Both sides of an equation will remain equivalent as long as we add, subtract, multiply and divide by equivalent expressions. The properties of equality ensure that we can isolate and solve for variables.

When two or more equations have the same solution, they are equivalent equations. For example, equations such as 2x – 4 = 0, 2x = 4, and x = 2 all have the same solution. When equivalent equations are substituted for one another, the form of the equation may change, but the meaning of the equation remains the same. It may just be in a form that is more easily solved.

The Addition Property of Equality in symbol form states that if a = b, then a + x = b + x. In other words, the same number can be added to both sides and the equations will still be equivalent. Therefore, 2x – 4 + 4 = 0 + 4, as the same number, 4, has been added to both sides. The equivalent equations are 2x = 4, as -4 + 4 equals 0 and can be eliminated from the first equation. The Addition Property of Equality also holds for subtraction, as the definition of subtraction is adding the inverse. Thus a – b is equivalent to a + (-b). The Addition Property of Equality can be used to isolate a variable on one side of the equation so that it can be solved. Suppose that y + 3 = 10. Then y + 3 -3 + 10 – 3, or y = 10 – 3, or y = 7. It is very important to solve for the variable by adding the same number to both sides of the equation, and to choose a number that will isolate the variable on one side of the equation. In the example, y + 3 = 10, the -3 isolated the variable y on the left side of the equation, and it was subtracted from both sides to keep the equations balanced.

Figure 1: The Addition Property of Equality in symbol form.

A reciprocal is the multiplicative inverse of any real number, so that the number times its reciprocal is equal to 1. In symbol form b ∙ 1/b =1 for any real number where b is not equal to 0. The number 0 has no reciprocal because division by 0 is undefined. For example, if b equals 100, 1/b equals 1/100, because 100 ∙ 1/100 is equal to 1. Similarly, if b equals 3/5, its reciprocal 1/b equals 5/3, because 3/5 times 5/3 is equal to 1.

Figure 2: A number multiplied by its reciprocal equals 1.

The Multiplication Property of Equality is similar to the Addition Property of Equality. In symbol form, if a =b, then ax = bx. The difference is that the Multiplication Property is used when the coefficient of the term is something other than 1. For example, if 2x = 4, the coefficient of x is 2. The Multiplication Property of Equality is also used to isolate a variable to one side of the equation so it can be solved. In order to isolate the x on the left side of the equation, both sides of the equation are multiplied by the reciprocal of 2, or ½ so that 2x ∙ 1/2 = 4 ∙ ½, or x = 2. In this example, both sides are multiplied by ½ to keep the equation balanced.

Figure 3: The Multiplication Property of Equality in symbol form.

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]]>Scientific notation is a method of representing very large numbers with positive exponents and very small numbers with negative exponents. Although any number can be used as a base to an exponent, scientific notation uses the decimal system, base 10.

The decimal system is based on powers of 10. The number 1 is 10^{0} and the number 10 is 10^{1}, by definition. The number 100 is 10^{2}, 1000 is 10^{3}, and 10000 is 10^{4}. The pattern is that the number of zeroes after the 1 is equal to the number represented by the exponent. Numbers that are smaller than 1 can be represented by negative exponents. A decimal such as 0.1, or 1/10, is represented with a negative exponent, 10^{-1}. Similarly, 0.01, or 1/100, is represented as 10^{-2}, 0.001 as 10^{-3}, and so on.

Figure 1: The decimal system is organized in powers of 10.

Scientific notation is an application of the rules about exponents, and are a way to express very large numbers (such as the distances to planets, stars, and galaxies) and very small numbers (such as the size of microbes, viruses, and atoms. Suppose the number is the distance from the earth to the sun, approximately 93 million miles or 93 ∙ 10^{6}. Using standard scientific notation, the number with significant digits is always expressed as a number between 1 and 10, or 9.3. Now 9.3 can be transformed to 93 by multiplying by 10^{1}, so that 9.3 ∙10^{1}∙10^{6} is 9.3 ∙10^{7}, also known as 93,000,000.

Figure 2: An example of standard scientific notation.

Significant digits for the purpose of scientific notation refer to the part of the number that is not included in the zeroes that follow it (in large numbers) or precede it (in small numbers). The significant digits in 93 million are the 9 and the 3. If the number were 92,900,000, the significant digits would be 929. If the number were 0.002135, the significant digits would be 2135.

Base 2, also known as the binary system, is used in computer applications, and can also be used in scientific notation. Engineering notation is a variation of scientific notation where the power of 10 that is used is a multiple of 1000. Suppose the number 83200 is expressed in engineering notation. It would be written as 83.2∙ 10^{3}, while in standard scientific notation it would be 8.32 ∙10^{4}.

Figure 3: Engineering notation is a variation of scientific notation.

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]]>Many applied problems can be solved using rational equations and proportions. Some of the types of problems that can be solved include problems involving different rates of work, different rates of motion, proportions, or similar triangles.

Many problems that involve rates of work are actually rational equations. Suppose that it takes Geordi 4 hours to realign the phase conduits and Barclay 6 hours to realign the phase conduits in the engine room of the Starship Enterprise. How long will it take them to do the job working together? Some fraction of 4, or, t/4, will be Geordi’s share of the work, and some fraction of 6, or t/6 will be Barclay’s share of the work, and both t/4 + t/6 will equal 1. Both fractions should be in like terms to add them, so 3t/12 +2t/12, or 5t/12 =1. Multiplying both sides by 12, 5t = 12, so t = 12/5 or 2 2/5 hours for the work in the engine room to be complete.

Figure 1: Work problems answer the question, When will the job be complete?

Most motion problems involve some form of the relationship between distance, rate and time, such that distance = rate ∙ time. Suppose that a Vulcan sehlat can run 20 km faster than a Klingon targ. The Vulcan sehlat can run 7 km in the time it takes the Klingon targ to run 5 km. How fast can both run? There are 2 equations that can be solved in this system; 5=rt for the Klingon targ, and 7 = (r + 20) t for the Vulcan sehlat. Solving each equation for t means that t = 5/r for the targ, and t = 7/(r + 20) for the sehlat. Since the times are the same for both animals, 5/r =7/(r + 20). Unlike the work problem, there are no factors in common. In order to put both fractions in like terms and cancel the fractions in the denominator, both sides of the equation can be multiplied by r(r + 20). Therefore 5(r + 20) = 7r, or 5r + 100 = 7r, or 100 = 7r – 5r or 100 = 2r, or r = 50. The targ runs 50 km an hour and the sehlat runs 50 + 20, or 70 km an hour.

Problems that involve comparable proportions use rational equations. Voyager is running out of food supplies, so an away team goes to an uncharted system in the Delta Quadrant. Neelix finds a species of edible fish in a lake. He has no idea how many there might be in the lake and doesn’t want to disturb the ecology of the lake too much. Harry Kim has a plan. If 300 fish are tagged and then set free, the proportion of the tagged fish that are caught to the total number of fish caught will be roughly the same proportion as the 300 fish that were tagged to the total number of fish in the lake. Harry and Neelix together catch 115 fish and 25 of them are tagged. (Since all 115 are edible, they transport them back to the starship.) So 25 is to 115 as what proportion of 300? The problem can be set up as 25/115 = 300/x. In order to put both in like terms and cancel the fractions in the denominator both sides can be multiplied by 115x. This leaves 25x = 115∙300 = 34500, so x equals 1380.

Figure 2: Proportion problems use rational equations.

Similar triangles are triangles with corresponding angles equal and corresponding sides proportional. On the planet Triskelion, the triangle is often used as a symbol. Although Captain Kirk was too busy fighting for his life to notice, the triangular medallion that the overseer was wearing and the triangular arena where he was fighting were similar triangles. If the medallion had two sides with measurements z tries and 5 tries and the triangular arena had corresponding sides with measurements 100 tries and 80 tries, what was the measurement of the side of the medallion? This problem is also a proportion, z/5 = 100/80, or 80z = 500, so z = 6.25.

Figure 3: Proportions of similar triangles can be solved by using rational expressions.

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