{"id":487,"date":"2024-04-19T20:46:25","date_gmt":"2024-04-19T20:46:25","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=487"},"modified":"2025-08-21T22:58:31","modified_gmt":"2025-08-21T22:58:31","slug":"introduction-to-real-numbers-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/introduction-to-real-numbers-learn-it-5\/","title":{"raw":"Introduction to Real Numbers: Learn It 5","rendered":"Introduction to Real Numbers: Learn It 5"},"content":{"raw":"

Algebraic Expressions<\/h2>\r\nNow that we've got the hang of using real numbers and the order of operations, let's step up our game! So far, the mathematical expressions we have seen have involved real numbers only.\r\n\r\nIn mathematics, we may see expressions such as [latex]x+5,\\frac{4}{3}\\pi {r}^{3}[\/latex], or [latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]. In the expression [latex]x+5, 5[\/latex] is called a constant<\/strong> because it does not vary and x<\/em> is called a variable<\/strong> because it does. An algebraic expression<\/strong> is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division. For example, [latex]3x + 2y - 7[\/latex] is an algebraic expression that contains two variables [latex]x[\/latex] and [latex]y[\/latex] and three constants [latex]3[\/latex], [latex]2[\/latex], and [latex]7[\/latex].\r\n\r\n
\r\n

constant, variable, algebraic expression<\/h3>\r\n
    \r\n \t
  • A constant<\/strong> is a fixed value or a number that does not change in a particular context.<\/li>\r\n \t
  • A variable<\/strong> is a symbol that represents a value or quantity that can change or vary in a given situation or context.<\/li>\r\n \t
  • An algebraic expression<\/strong> is a mathematical phrase or combination of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division.<\/li>\r\n<\/ul>\r\n<\/section>\r\n

    Evaluating Algebraic Expressions<\/h3>\r\nEvaluation of an algebraic expression involves a specific process where we replace each variable in the expression with a given number, and then calculate the result using the established order of operations.\r\n\r\n
    How To: Evaluate Algebraic Expressions<\/strong>Use the following steps to evaluate an algebraic expression:\r\n
      \r\n \t
    1. Replace each variable in the expression with the given value<\/li>\r\n \t
    2. Simplify the resulting expression using the order of operations<\/li>\r\n<\/ol>\r\nNote: If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.\r\n\r\n<\/section>
      Evaluate the expression [latex]2x - 7[\/latex] for each value for [latex]x=1[\/latex].<\/em>[reveal-answer q=\"789609\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"789609\"]Here's how it works:\r\n
        \r\n \t
      1. Substitute the Variable<\/strong>: Replace [latex]x[\/latex] with [latex]1[\/latex] in the expression, turning [latex]2x - 7[\/latex] into [latex]2(1)-7[\/latex]<\/li>\r\n \t
      2. Perform Calculations<\/strong>: Next, follow the order of operations. First, multiply [latex]2[\/latex] by [latex]1[\/latex]to get [latex]2[\/latex], and then subtract [latex]7[\/latex] to get [latex]-5[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
        [ohm2_question hide_question_numbers=1]18666[\/ohm2_question]<\/section>
        Evaluate each expression for the given values.\r\n
          \r\n \t
        1. [latex]x+5[\/latex] for [latex]x=-5[\/latex]<\/li>\r\n \t
        2. [latex]\\frac{t}{2t - 1}[\/latex] for [latex]t=10[\/latex]<\/li>\r\n \t
        3. [latex]\\dfrac{4}{3}\\pi {r}^{3}[\/latex] for [latex]r=5[\/latex]<\/li>\r\n \t
        4. [latex]a+ab+b[\/latex] for [latex]a=11,b=-8[\/latex]<\/li>\r\n \t
        5. [latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex] for [latex]m=2,n=3[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"182854\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"182854\"]\r\n
            \r\n \t
          1. Substitute [latex]-5[\/latex] for [latex]x[\/latex].\r\n
            [latex]\\begin{align}x+5 &=\\left(-5\\right)+5 \\\\ &=0\\end{align}[\/latex]<\/div><\/li>\r\n \t
          2. Substitute 10 for [latex]t[\/latex].\r\n
            [latex]\\begin{align}\\frac{t}{2t-1} & =\\frac{\\left(10\\right)}{2\\left(10\\right)-1} \\\\ & =\\frac{10}{20-1} \\\\ & =\\frac{10}{19}\\end{align}[\/latex]<\/div><\/li>\r\n \t
          3. Substitute 5 for [latex]r[\/latex].\r\n
            [latex]\\begin{align}\\frac{4}{3}\\pi r^{3} & =\\frac{4}{3}\\pi\\left(5\\right)^{3} \\\\ & =\\frac{4}{3}\\pi\\left(125\\right) \\\\ & =\\frac{500}{3}\\pi\\end{align}[\/latex]<\/div><\/li>\r\n \t
          4. Substitute 11 for [latex]a[\/latex] and \u20138 for [latex]b[\/latex].\r\n
            [latex]\\begin{align}a+ab+b & =\\left(11\\right)+\\left(11\\right)\\left(-8\\right)+\\left(-8\\right) \\\\ & =11-8-8 \\\\ & =-85\\end{align}[\/latex]<\/div><\/li>\r\n \t
          5. Substitute 2 for [latex]m[\/latex] and 3 for [latex]n[\/latex].\r\n
            [latex]\\begin{align}\\sqrt{2m^{3}n^{2}} & =\\sqrt{2\\left(2\\right)^{3}\\left(3\\right)^{2}} \\\\ & =\\sqrt{2\\left(8\\right)\\left(9\\right)} \\\\ & =\\sqrt{144} \\\\ & =12\\end{align}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
            [ohm2_question hide_question_numbers=1]6621[\/ohm2_question]<\/section>
            [ohm2_question hide_question_numbers=1]6622[\/ohm2_question]<\/section>
            [ohm2_question hide_question_numbers=1]6753[\/ohm2_question]<\/section>\r\n

            Algebraic Formulas<\/h2>\r\nAn equation<\/strong> is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation [latex]2x+1=7[\/latex] has the unique solution [latex]x=3[\/latex] because when we substitute [latex]3[\/latex] for [latex]x[\/latex] in the equation, we obtain the true statement [latex]2\\left(3\\right)+1=7[\/latex].\r\n\r\nA formula<\/strong> is an equation expressing a relationship between constant and variable quantities. Very often the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area [latex]A[\/latex] of a circle in terms of the radius [latex]r[\/latex] of the circle: [latex]A=\\pi {r}^{2}[\/latex]. For any value of [latex]r[\/latex], the area [latex]A[\/latex] can be found by evaluating the expression [latex]\\pi {r}^{2}[\/latex].\r\n\r\n
            \r\n
            \r\n

            Equations and formulas<\/h3>\r\n
              \r\n \t
            • An equation<\/strong> is a mathematical statement that shows the equality of two expressions, typically separated by an equal sign. It states that the two expressions have the same value, and the values of variables that make the equation true are called solutions.<\/li>\r\n \t
            • A formula<\/strong> is a mathematical expression that represents a relationship or a rule between variables or quantities. It usually contains variables, constants, and arithmetic operations, and is used to calculate or derive a particular result or value.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>
              A right circular cylinder with radius [latex]r[\/latex] and height [latex]h[\/latex] has the surface area [latex]S[\/latex] (in square units) given by the formula [latex]S=2\\pi r\\left(r+h\\right)[\/latex]. Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of [latex]\\pi[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"A Right circular cylinder with radius and height labeled[\/caption]\r\n\r\n[reveal-answer q=\"257174\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"257174\"]\r\n\r\nEvaluate the expression [latex]2\\pi r\\left(r+h\\right)[\/latex] for [latex]r=6[\/latex] and [latex]h=9[\/latex].\r\n
              [latex]\\begin{align}S&=2\\pi r\\left(r+h\\right) \\\\ & =2\\pi\\left(6\\right)[\\left(6\\right)+\\left(9\\right)] \\\\ & =2\\pi\\left(6\\right)\\left(15\\right) \\\\ & =180\\pi\\end{align}[\/latex]<\/div>\r\n \r\n\r\nThe surface area is [latex]180\\pi [\/latex] square inches.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
              [ohm2_question hide_question_numbers=1]18717[\/ohm2_question]<\/section><\/section>","rendered":"

              Algebraic Expressions<\/h2>\n

              Now that we’ve got the hang of using real numbers and the order of operations, let’s step up our game! So far, the mathematical expressions we have seen have involved real numbers only.<\/p>\n

              In mathematics, we may see expressions such as [latex]x+5,\\frac{4}{3}\\pi {r}^{3}[\/latex], or [latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]. In the expression [latex]x+5, 5[\/latex] is called a constant<\/strong> because it does not vary and x<\/em> is called a variable<\/strong> because it does. An algebraic expression<\/strong> is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division. For example, [latex]3x + 2y - 7[\/latex] is an algebraic expression that contains two variables [latex]x[\/latex] and [latex]y[\/latex] and three constants [latex]3[\/latex], [latex]2[\/latex], and [latex]7[\/latex].<\/p>\n

              \n

              constant, variable, algebraic expression<\/h3>\n
                \n
              • A constant<\/strong> is a fixed value or a number that does not change in a particular context.<\/li>\n
              • A variable<\/strong> is a symbol that represents a value or quantity that can change or vary in a given situation or context.<\/li>\n
              • An algebraic expression<\/strong> is a mathematical phrase or combination of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division.<\/li>\n<\/ul>\n<\/section>\n

                Evaluating Algebraic Expressions<\/h3>\n

                Evaluation of an algebraic expression involves a specific process where we replace each variable in the expression with a given number, and then calculate the result using the established order of operations.<\/p>\n

                How To: Evaluate Algebraic Expressions<\/strong>Use the following steps to evaluate an algebraic expression:<\/p>\n
                  \n
                1. Replace each variable in the expression with the given value<\/li>\n
                2. Simplify the resulting expression using the order of operations<\/li>\n<\/ol>\n

                  Note: If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.<\/p>\n<\/section>\n

                  Evaluate the expression [latex]2x - 7[\/latex] for each value for [latex]x=1[\/latex].<\/em><\/p>\n