{"id":3523,"date":"2024-09-06T11:21:09","date_gmt":"2024-09-06T11:21:09","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=3523"},"modified":"2025-08-13T15:51:12","modified_gmt":"2025-08-13T15:51:12","slug":"non-linear-equations-and-inequalities-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/non-linear-equations-and-inequalities-get-stronger\/","title":{"raw":"Non-Linear Equations Get Stronger","rendered":"Non-Linear Equations Get Stronger"},"content":{"raw":"

Quadratic Equations<\/span><\/h2>\r\n

For the following exercises, solve the quadratic equation by factoring.<\/p>\r\n\r\n

    \r\n \t
  1. [latex]x^2 - 9x + 18 = 0[\/latex]<\/li>\r\n \t
  2. [latex]6x^2 + 17x + 5 = 0[\/latex]<\/li>\r\n \t
  3. [latex]3x^2 - 75 = 0[\/latex]<\/li>\r\n \t
  4. [latex]4x^2 = 9[\/latex]<\/li>\r\n \t
  5. [latex]5x^2 = 5x + 30[\/latex]<\/li>\r\n \t
  6. [latex]7x^2 + 3x = 0[\/latex]<\/li>\r\n<\/ol>\r\n

    For the following exercises, solve the quadratic equation by using the square root property.<\/p>\r\n\r\n

      \r\n \t
    1. [latex]x^2 = 36[\/latex]<\/li>\r\n \t
    2. [latex](x-1)^2 = 25[\/latex]<\/li>\r\n \t
    3. [latex](2x+1)^2 = 9[\/latex]<\/li>\r\n<\/ol>\r\n

      For the following exercises, solve the quadratic equation by completing the square. Show each step.<\/p>\r\n\r\n

        \r\n \t
      1. [latex]x^2 - 9x - 22 = 0[\/latex]<\/li>\r\n \t
      2. [latex]x^2 - 6x = 13[\/latex]<\/li>\r\n \t
      3. [latex]2 + z = 6z^2[\/latex]<\/li>\r\n \t
      4. [latex]2x^2 - 3x - 1 = 0[\/latex]<\/li>\r\n<\/ol>\r\n

        For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.<\/p>\r\n\r\n

          \r\n \t
        1. [latex]x^2 + 4x + 7 = 0[\/latex]<\/li>\r\n \t
        2. [latex]9x^2 - 30x + 25 = 0[\/latex]<\/li>\r\n \t
        3. [latex]6x^2 - x - 2 = 0[\/latex]<\/li>\r\n<\/ol>\r\n

          For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.<\/p>\r\n\r\n

            \r\n \t
          1. [latex]x^2 + x = 4[\/latex]<\/li>\r\n \t
          2. [latex]3x^2 - 5x + 1 = 0[\/latex]<\/li>\r\n \t
          3. [latex]4 + \\dfrac{1}{x} - \\dfrac{1}{x^2} = 0[\/latex]<\/li>\r\n<\/ol>\r\n

            Extensions<\/p>\r\n\r\n

              \r\n \t
            1. Beginning with the general form of a quadratic equation, [latex]ax^2 + bx + c = 0[\/latex], solve for x by using the completing the square method, thus deriving the quadratic formula.<\/li>\r\n \t
            2. A person has a garden that has a length [latex]10[\/latex] feet longer than the width. Set up a quadratic equation to find the dimensions of the garden if its area is [latex]119[\/latex] ft[latex]^2[\/latex]. Solve the quadratic equation to find the length and width.<\/li>\r\n \t
            3. Suppose that an equation is given [latex]p = -2x^2 + 280x - 1000[\/latex], where x represents the number of items sold at an auction and p is the profit made by the business that ran the auction. How many items sold would make this profit a maximum? Solve this by graphing the expression in your graphing utility and finding the maximum using 2[latex]^{\\text{nd}}[\/latex] CALC maximum. To obtain a good window for the curve, set [latex]x[\/latex] [latex][0,200] [\/latex]and [latex]y[\/latex] [latex][0,10000][\/latex].<\/li>\r\n<\/ol>\r\n

              Real-World Applications<\/p>\r\n\r\n

                \r\n \t
              1. The cost function for a certain company is [latex]C = 60x + 300[\/latex] and the revenue is given by [latex]R = 100x - 0.5x^2[\/latex]. Recall that profit is revenue minus cost. Set up a quadratic equation and find two values of [latex]x[\/latex] (production level) that will create a profit of [latex]$300[\/latex].<\/li>\r\n \t
              2. A vacant lot is being converted into a community garden. The garden and the walkway around its perimeter have an area of [latex]378[\/latex] ft[latex]^2[\/latex]. Find the width of the walkway if the garden is [latex]12[\/latex] ft. wide by [latex]15[\/latex] ft. long.\r\n\r\n[caption id=\"attachment_6012\" align=\"alignnone\" width=\"378\"]\"A Lot enclosed by walls with labels[\/caption]<\/li>\r\n<\/ol>\r\n

                Other Types of Equations<\/span><\/h2>\r\n

                For the following exercises, solve each rational equation for x. State all x-values that are excluded from the solution set.<\/p>\r\n\r\n

                  \r\n \t
                1. [latex]\\dfrac{3}{x} - \\dfrac{1}{3} = \\dfrac{1}{6}[\/latex]<\/li>\r\n \t
                2. [latex]\\dfrac{3}{x-2} - \\dfrac{1}{x-1} + \\dfrac{7}{(x-1)(x-2)}[\/latex]<\/li>\r\n \t
                3. [latex]\\dfrac{1}{x} = \\dfrac{1}{5} + \\dfrac{3}{2x}[\/latex]<\/li>\r\n<\/ol>\r\n

                  For the following exercises, solve the rational exponent equation. Use factoring where necessary.<\/p>\r\n\r\n

                    \r\n \t
                  1. [latex]x^{\\dfrac{3}{4}} = 27[\/latex]<\/li>\r\n \t
                  2. [latex](x-1)^{\\dfrac{3}{4}} = 8[\/latex]<\/li>\r\n \t
                  3. [latex]x^{\\dfrac{2}{3}} - 5x^{\\dfrac{1}{3}} + 6 = 0[\/latex]<\/li>\r\n<\/ol>\r\n

                    For the following exercises, solve the following polynomial equations by grouping and factoring.<\/p>\r\n\r\n

                      \r\n \t
                    1. [latex]x^3 + 2x^2 - x - 2 = 0[\/latex]<\/li>\r\n \t
                    2. [latex]4y^3 - 9y = 0[\/latex]<\/li>\r\n \t
                    3. [latex]m^3 + m^2 - m - 1 = 0[\/latex]<\/li>\r\n \t
                    4. [latex]5x^3 + 45x = 2x^2 + 18[\/latex]<\/li>\r\n<\/ol>\r\n

                      For the following exercises, solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions.<\/p>\r\n\r\n

                        \r\n \t
                      1. [latex]\\sqrt{x - 7} = 5[\/latex]<\/li>\r\n \t
                      2. [latex]\\sqrt{3t + 5} = 7[\/latex]<\/li>\r\n \t
                      3. [latex]\\sqrt{12 - x} = x[\/latex]<\/li>\r\n \t
                      4. [latex]\\sqrt{3x + 7} + \\sqrt{x + 2} = 1[\/latex]<\/li>\r\n<\/ol>\r\n

                        For the following exercises, solve the equation involving absolute value.<\/p>\r\n\r\n

                          \r\n \t
                        1. [latex]|3x - 4| = 8[\/latex]<\/li>\r\n \t
                        2. [latex]|1 - 4x| - 1 = 5[\/latex]<\/li>\r\n \t
                        3. [latex]|2x - 1| - 7 = -2[\/latex]<\/li>\r\n \t
                        4. [latex]|x + 5| = 0[\/latex]<\/li>\r\n<\/ol>\r\n

                          For the following exercises, use the model for the period of a pendulum, [latex]T[\/latex], such that [latex]T = 2\\pi\\sqrt{\\dfrac{L}{g}}[\/latex], where the length of the pendulum is [latex]L[\/latex] and the acceleration due to gravity is [latex]g[\/latex].<\/p>\r\n\r\n

                            \r\n \t
                          1. If the acceleration due to gravity is [latex]9.8[\/latex] m\/s[latex]^2[\/latex] and the period equals [latex]1[\/latex] s, find the length to the nearest cm ([latex]100[\/latex] cm = [latex]1[\/latex] m).<\/li>\r\n \t
                          2. If the gravity is [latex]32[\/latex] ft\/s[latex]^2[\/latex] and the period equals [latex]1[\/latex] s, find the length to the nearest in. ([latex]12[\/latex] in. = [latex]1[\/latex] ft). Round your answer to the nearest in.<\/li>\r\n<\/ol>\r\n

                            For the following exercises, use a model for body surface area, BSA, such that [latex]BSA = \\sqrt{\\dfrac{wh}{3600}}[\/latex], where [latex]w[\/latex] = weight in kg and [latex]h[\/latex] = height in cm.<\/p>\r\n\r\n

                              \r\n \t
                            1. Find the height of a [latex]72[\/latex]-kg female to the nearest cm whose [latex]BSA = 1.8[\/latex].<\/li>\r\n \t
                            2. Find the weight of a [latex]177[\/latex]-cm male to the nearest kg whose [latex]BSA = 2.1[\/latex].<\/li>\r\n<\/ol>\r\n

                              For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation.<\/p>\r\n\r\n

                                \r\n \t
                              1. [latex]|x + 9| \\geq -6[\/latex]<\/li>\r\n \t
                              2. [latex]|3x - 1| > 11[\/latex]<\/li>\r\n \t
                              3. [latex]|x - 2| + 4 \\geq 10[\/latex]<\/li>\r\n \t
                              4. [latex]|x - 7| < -4[\/latex]<\/li>\r\n \t
                              5. [latex]|\\dfrac{x-3}{4}| < 2[\/latex]<\/li>\r\n<\/ol>","rendered":"

                                Quadratic Equations<\/span><\/h2>\n

                                For the following exercises, solve the quadratic equation by factoring.<\/p>\n

                                  \n
                                1. [latex]x^2 - 9x + 18 = 0[\/latex]<\/li>\n
                                2. [latex]6x^2 + 17x + 5 = 0[\/latex]<\/li>\n
                                3. [latex]3x^2 - 75 = 0[\/latex]<\/li>\n
                                4. [latex]4x^2 = 9[\/latex]<\/li>\n
                                5. [latex]5x^2 = 5x + 30[\/latex]<\/li>\n
                                6. [latex]7x^2 + 3x = 0[\/latex]<\/li>\n<\/ol>\n

                                  For the following exercises, solve the quadratic equation by using the square root property.<\/p>\n

                                    \n
                                  1. [latex]x^2 = 36[\/latex]<\/li>\n
                                  2. [latex](x-1)^2 = 25[\/latex]<\/li>\n
                                  3. [latex](2x+1)^2 = 9[\/latex]<\/li>\n<\/ol>\n

                                    For the following exercises, solve the quadratic equation by completing the square. Show each step.<\/p>\n

                                      \n
                                    1. [latex]x^2 - 9x - 22 = 0[\/latex]<\/li>\n
                                    2. [latex]x^2 - 6x = 13[\/latex]<\/li>\n
                                    3. [latex]2 + z = 6z^2[\/latex]<\/li>\n
                                    4. [latex]2x^2 - 3x - 1 = 0[\/latex]<\/li>\n<\/ol>\n

                                      For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.<\/p>\n

                                        \n
                                      1. [latex]x^2 + 4x + 7 = 0[\/latex]<\/li>\n
                                      2. [latex]9x^2 - 30x + 25 = 0[\/latex]<\/li>\n
                                      3. [latex]6x^2 - x - 2 = 0[\/latex]<\/li>\n<\/ol>\n

                                        For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.<\/p>\n

                                          \n
                                        1. [latex]x^2 + x = 4[\/latex]<\/li>\n
                                        2. [latex]3x^2 - 5x + 1 = 0[\/latex]<\/li>\n
                                        3. [latex]4 + \\dfrac{1}{x} - \\dfrac{1}{x^2} = 0[\/latex]<\/li>\n<\/ol>\n

                                          Extensions<\/p>\n

                                            \n
                                          1. Beginning with the general form of a quadratic equation, [latex]ax^2 + bx + c = 0[\/latex], solve for x by using the completing the square method, thus deriving the quadratic formula.<\/li>\n
                                          2. A person has a garden that has a length [latex]10[\/latex] feet longer than the width. Set up a quadratic equation to find the dimensions of the garden if its area is [latex]119[\/latex] ft[latex]^2[\/latex]. Solve the quadratic equation to find the length and width.<\/li>\n
                                          3. Suppose that an equation is given [latex]p = -2x^2 + 280x - 1000[\/latex], where x represents the number of items sold at an auction and p is the profit made by the business that ran the auction. How many items sold would make this profit a maximum? Solve this by graphing the expression in your graphing utility and finding the maximum using 2[latex]^{\\text{nd}}[\/latex] CALC maximum. To obtain a good window for the curve, set [latex]x[\/latex] [latex][0,200][\/latex]and [latex]y[\/latex] [latex][0,10000][\/latex].<\/li>\n<\/ol>\n

                                            Real-World Applications<\/p>\n

                                              \n
                                            1. The cost function for a certain company is [latex]C = 60x + 300[\/latex] and the revenue is given by [latex]R = 100x - 0.5x^2[\/latex]. Recall that profit is revenue minus cost. Set up a quadratic equation and find two values of [latex]x[\/latex] (production level) that will create a profit of [latex]$300[\/latex].<\/li>\n
                                            2. A vacant lot is being converted into a community garden. The garden and the walkway around its perimeter have an area of [latex]378[\/latex] ft[latex]^2[\/latex]. Find the width of the walkway if the garden is [latex]12[\/latex] ft. wide by [latex]15[\/latex] ft. long.
                                              \n
                                              \"A
                                              Lot enclosed by walls with labels<\/figcaption><\/figure>\n<\/li>\n<\/ol>\n

                                              Other Types of Equations<\/span><\/h2>\n

                                              For the following exercises, solve each rational equation for x. State all x-values that are excluded from the solution set.<\/p>\n

                                                \n
                                              1. [latex]\\dfrac{3}{x} - \\dfrac{1}{3} = \\dfrac{1}{6}[\/latex]<\/li>\n
                                              2. [latex]\\dfrac{3}{x-2} - \\dfrac{1}{x-1} + \\dfrac{7}{(x-1)(x-2)}[\/latex]<\/li>\n
                                              3. [latex]\\dfrac{1}{x} = \\dfrac{1}{5} + \\dfrac{3}{2x}[\/latex]<\/li>\n<\/ol>\n

                                                For the following exercises, solve the rational exponent equation. Use factoring where necessary.<\/p>\n

                                                  \n
                                                1. [latex]x^{\\dfrac{3}{4}} = 27[\/latex]<\/li>\n
                                                2. [latex](x-1)^{\\dfrac{3}{4}} = 8[\/latex]<\/li>\n
                                                3. [latex]x^{\\dfrac{2}{3}} - 5x^{\\dfrac{1}{3}} + 6 = 0[\/latex]<\/li>\n<\/ol>\n

                                                  For the following exercises, solve the following polynomial equations by grouping and factoring.<\/p>\n

                                                    \n
                                                  1. [latex]x^3 + 2x^2 - x - 2 = 0[\/latex]<\/li>\n
                                                  2. [latex]4y^3 - 9y = 0[\/latex]<\/li>\n
                                                  3. [latex]m^3 + m^2 - m - 1 = 0[\/latex]<\/li>\n
                                                  4. [latex]5x^3 + 45x = 2x^2 + 18[\/latex]<\/li>\n<\/ol>\n

                                                    For the following exercises, solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions.<\/p>\n

                                                      \n
                                                    1. [latex]\\sqrt{x - 7} = 5[\/latex]<\/li>\n
                                                    2. [latex]\\sqrt{3t + 5} = 7[\/latex]<\/li>\n
                                                    3. [latex]\\sqrt{12 - x} = x[\/latex]<\/li>\n
                                                    4. [latex]\\sqrt{3x + 7} + \\sqrt{x + 2} = 1[\/latex]<\/li>\n<\/ol>\n

                                                      For the following exercises, solve the equation involving absolute value.<\/p>\n

                                                        \n
                                                      1. [latex]|3x - 4| = 8[\/latex]<\/li>\n
                                                      2. [latex]|1 - 4x| - 1 = 5[\/latex]<\/li>\n
                                                      3. [latex]|2x - 1| - 7 = -2[\/latex]<\/li>\n
                                                      4. [latex]|x + 5| = 0[\/latex]<\/li>\n<\/ol>\n

                                                        For the following exercises, use the model for the period of a pendulum, [latex]T[\/latex], such that [latex]T = 2\\pi\\sqrt{\\dfrac{L}{g}}[\/latex], where the length of the pendulum is [latex]L[\/latex] and the acceleration due to gravity is [latex]g[\/latex].<\/p>\n

                                                          \n
                                                        1. If the acceleration due to gravity is [latex]9.8[\/latex] m\/s[latex]^2[\/latex] and the period equals [latex]1[\/latex] s, find the length to the nearest cm ([latex]100[\/latex] cm = [latex]1[\/latex] m).<\/li>\n
                                                        2. If the gravity is [latex]32[\/latex] ft\/s[latex]^2[\/latex] and the period equals [latex]1[\/latex] s, find the length to the nearest in. ([latex]12[\/latex] in. = [latex]1[\/latex] ft). Round your answer to the nearest in.<\/li>\n<\/ol>\n

                                                          For the following exercises, use a model for body surface area, BSA, such that [latex]BSA = \\sqrt{\\dfrac{wh}{3600}}[\/latex], where [latex]w[\/latex] = weight in kg and [latex]h[\/latex] = height in cm.<\/p>\n

                                                            \n
                                                          1. Find the height of a [latex]72[\/latex]-kg female to the nearest cm whose [latex]BSA = 1.8[\/latex].<\/li>\n
                                                          2. Find the weight of a [latex]177[\/latex]-cm male to the nearest kg whose [latex]BSA = 2.1[\/latex].<\/li>\n<\/ol>\n

                                                            For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation.<\/p>\n

                                                              \n
                                                            1. [latex]|x + 9| \\geq -6[\/latex]<\/li>\n
                                                            2. [latex]|3x - 1| > 11[\/latex]<\/li>\n
                                                            3. [latex]|x - 2| + 4 \\geq 10[\/latex]<\/li>\n
                                                            4. [latex]|x - 7| < -4[\/latex]<\/li>\n
                                                            5. [latex]|\\dfrac{x-3}{4}| < 2[\/latex]<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":24,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":92,"module-header":"practice","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/3523"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":13,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/3523\/revisions"}],"predecessor-version":[{"id":7621,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/3523\/revisions\/7621"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/92"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/3523\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=3523"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=3523"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=3523"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=3523"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}