{"id":320,"date":"2026-02-02T19:36:45","date_gmt":"2026-02-02T19:36:45","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/?post_type=chapter&#038;p=320"},"modified":"2026-02-11T18:49:19","modified_gmt":"2026-02-11T18:49:19","slug":"matrices-and-matrix-operations-get-stronger-answer-key","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/matrices-and-matrix-operations-get-stronger-answer-key\/","title":{"raw":"Systems of Equations and Inequalities: Get Stronger Answer Key","rendered":"Systems of Equations and Inequalities: Get Stronger Answer Key"},"content":{"raw":"<h2>Systems of Linear Equations: Two Variables<\/h2>\r\n1.\u00a0No, you can either have zero, one, or infinitely many. Examine graphs.\r\n\r\n5.\u00a0You can solve by substitution (isolating [latex]x[\/latex] or [latex]y[\/latex] ), graphically, or by addition.\r\n\r\n7.\u00a0Yes\r\n\r\n9.\u00a0Yes\r\n\r\n11.\u00a0[latex]\\left(-1,2\\right)[\/latex]\r\n\r\n13.\u00a0[latex]\\left(-3,1\\right)[\/latex]\r\n\r\n15.\u00a0[latex]\\left(-\\frac{3}{5},0\\right)[\/latex]\r\n\r\n17.\u00a0No solutions exist.\r\n\r\n19.\u00a0[latex]\\left(\\frac{72}{5},\\frac{132}{5}\\right)[\/latex]\r\n\r\n21.\u00a0[latex]\\left(6,-6\\right)[\/latex]\r\n\r\n23.\u00a0[latex]\\left(-\\frac{1}{2},\\frac{1}{10}\\right)[\/latex]\r\n\r\n25.\u00a0No solutions exist.\r\n\r\n27.\u00a0[latex]\\left(-\\frac{1}{5},\\frac{2}{3}\\right)[\/latex]\r\n\r\n29.\u00a0[latex]\\left(x,\\frac{x+3}{2}\\right)[\/latex]\r\n\r\n31.\u00a0[latex]\\left(-4,4\\right)[\/latex]\r\n\r\n33.\u00a0[latex]\\left(\\frac{1}{2},\\frac{1}{8}\\right)[\/latex]\r\n\r\n35.\u00a0[latex]\\left(\\frac{1}{6},0\\right)[\/latex]\r\n\r\n37.\u00a0[latex]\\left(x,2\\left(7x - 6\\right)\\right)[\/latex]\r\n\r\n39.\u00a0[latex]\\left(-\\frac{5}{6},\\frac{4}{3}\\right)[\/latex]\r\n\r\n41.\u00a0Consistent with one solution\r\n\r\n43.\u00a0Consistent with one solution\r\n\r\n45.\u00a0Dependent with infinitely many solutions\r\n\r\n47.\u00a0[latex]\\left(-3.08,4.91\\right)[\/latex]\r\n\r\n49.\u00a0[latex]\\left(-1.52,2.29\\right)[\/latex]\r\n\r\n61.\u00a0The numbers are 7.5 and 20.5.\r\n\r\n63.\u00a024,000\r\n\r\n65.\u00a0790 sophomores, 805 freshman\r\n\r\n67.\u00a056 men, 74 women\r\n\r\n69.\u00a010 gallons of 10% solution, 15 gallons of 60% solution\r\n\r\n73.\u00a0$12,500 in the first account, $10,500 in the second account.\r\n<h2>Systems of Linear Equations: Three Variables<\/h2>\r\n1.\u00a0No, there can be only one, zero, or infinitely many solutions.\r\n\r\n7. No\r\n\r\n9.\u00a0Yes\r\n\r\n11.\u00a0[latex]\\left(-1,4,2\\right)[\/latex]\r\n\r\n13.\u00a0[latex]\\left(-\\frac{85}{107},\\frac{312}{107},\\frac{191}{107}\\right)[\/latex]\r\n\r\n15.\u00a0[latex]\\left(1,\\frac{1}{2},0\\right)[\/latex]\r\n\r\n17.\u00a0[latex]\\left(4,-6,1\\right)[\/latex]\r\n\r\n19.\u00a0[latex]\\left(x,\\frac{1}{27}\\left(65 - 16x\\right),\\frac{x+28}{27}\\right)[\/latex]\r\n\r\n21.\u00a0[latex]\\left(-\\frac{45}{13},\\frac{17}{13},-2\\right)[\/latex]\r\n\r\n23.\u00a0No solutions exist\r\n\r\n25.\u00a0[latex]\\left(0,0,0\\right)[\/latex]\r\n\r\n27.\u00a0[latex]\\left(\\frac{4}{7},-\\frac{1}{7},-\\frac{3}{7}\\right)[\/latex]\r\n\r\n29.\u00a0[latex]\\left(7,20,16\\right)[\/latex]\r\n\r\n39.\u00a0[latex]\\left(\\frac{1}{2},\\frac{1}{5},\\frac{4}{5}\\right)[\/latex]\r\n\r\n43.\u00a0[latex]\\left(2,0,0\\right)[\/latex]\r\n\r\n51.\u00a024, 36, 48\r\n\r\n53.\u00a070 grandparents, 140 parents, 190 children\r\n\r\n55.\u00a0Your share was $19.95, Sarah\u2019s share was $40, and your other roommate\u2019s share was $22.05.\r\n\r\n57.\u00a0There are infinitely many solutions; we need more information\r\n\r\n59.\u00a0500 students, 225 children, and 450 adults\r\n\r\n63.\u00a0$400,000 in the account that pays 3% interest, $500,000 in the account that pays 4% interest, and $100,000 in the account that pays 2% interest.\r\n\r\n65.\u00a0The United States consumed 26.3%, Japan 7.1%, and China 6.4% of the world\u2019s oil.\r\n\r\n69.\u00a0Birds were 19.3%, fish were 18.6%, and mammals were 17.1% of endangered species\r\n<h2>Systems of Nonlinear Equations and Inequalities: Two Variables<\/h2>\r\n<h2>Solutions to Odd-Numbered Exercises<\/h2>\r\n3.\u00a0No. There does not need to be a feasible region. Consider a system that is bounded by two parallel lines. One inequality represents the region above the upper line; the other represents the region below the lower line. In this case, no points in the plane are located in both regions; hence there is no feasible region.\r\n\r\n7.\u00a0[latex]\\left(0,-3\\right),\\left(3,0\\right)[\/latex]\r\n\r\n9.\u00a0[latex]\\left(-\\frac{3\\sqrt{2}}{2},\\frac{3\\sqrt{2}}{2}\\right),\\left(\\frac{3\\sqrt{2}}{2},-\\frac{3\\sqrt{2}}{2}\\right)[\/latex]\r\n\r\n11.\u00a0[latex]\\left(-3,0\\right),\\left(3,0\\right)[\/latex]\r\n\r\n13.\u00a0[latex]\\left(\\frac{1}{4},-\\frac{\\sqrt{62}}{8}\\right),\\left(\\frac{1}{4},\\frac{\\sqrt{62}}{8}\\right)[\/latex]\r\n\r\n15.\u00a0[latex]\\left(-\\frac{\\sqrt{398}}{4},\\frac{199}{4}\\right),\\left(\\frac{\\sqrt{398}}{4},\\frac{199}{4}\\right)[\/latex]\r\n\r\n17.\u00a0[latex]\\left(0,2\\right),\\left(1,3\\right)[\/latex]\r\n\r\n19.\u00a0[latex]\\left(-\\sqrt{\\frac{1}{2}\\left(\\sqrt{5}-1\\right)},\\frac{1}{2}\\left(1-\\sqrt{5}\\right)\\right),\\left(\\sqrt{\\frac{1}{2}\\left(\\sqrt{5}-1\\right)},\\frac{1}{2}\\left(1-\\sqrt{5}\\right)\\right)[\/latex]\r\n\r\n21.\u00a0[latex]\\left(5,0\\right)[\/latex]\r\n\r\n25.\u00a0[latex]\\left(3,0\\right)[\/latex]\r\n\r\n27.\u00a0No Solutions Exist\r\n\r\n33.\u00a0[latex]\\left(2,0\\right)[\/latex]\r\n\r\n35.\u00a0[latex]\\left(-\\sqrt{7},-3\\right),\\left(-\\sqrt{7},3\\right),\\left(\\sqrt{7},-3\\right),\\left(\\sqrt{7},3\\right)[\/latex]\r\n\r\n41.\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181409\/CNX_Precalc_Figure_09_03_2042.jpg\" alt=\"A shaded figure with a dotted line that has two marked points. The first point is at square root of two minus 1, two times (the square root of two minus one). The second point is at negative one minus square root of two, negative two times (one plus the square root of two).\" width=\"487\" height=\"317\" \/>\r\n\r\n43.\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181412\/CNX_Precalc_Figure_09_03_2072.jpg\" alt=\"Two dotted, shaded figures with points marked. The first point is (negative square root of 37 over 2, 3 times square root of seven over two). The second point is (square root of 37 over 2, 3 times square root of 7 over two). The third point is (negative square root 37 over 2, negative 3 times square root 7 divided by 2). The fourth point is (square root 37 over 2, negative 3 times square root of 7 over two).\" width=\"487\" height=\"379\" \/>\r\n\r\n45.\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181415\/CNX_Precalc_Figure_09_03_2092.jpg\" alt=\"Two dotted, shaded figures with marked points. The first point is negative square root of nineteen-tenths, square root of forty-seven-tenths. The second point is square root of 19 tenths, square root of 47 tenths. The third point is negative square root of 19 tenths, negative square root of 47 tenths. The fourth point is square root of 19 tenths, negative square root of 47 tenths.\" width=\"487\" height=\"380\" \/>\r\n\r\n49.\u00a0[latex]\\left(-2\\sqrt{\\frac{70}{383}},-2\\sqrt{\\frac{35}{29}}\\right),\\left(-2\\sqrt{\\frac{70}{383}},2\\sqrt{\\frac{35}{29}}\\right),\\left(2\\sqrt{\\frac{70}{383}},-2\\sqrt{\\frac{35}{29}}\\right),\\left(2\\sqrt{\\frac{70}{383}},2\\sqrt{\\frac{35}{29}}\\right)[\/latex]\r\n\r\n51.\u00a0No Solution Exists\r\n\r\n55.\u00a012, 288\r\n\r\n57.\u00a02\u201320 computers\r\n<h2>Partial Fractions<\/h2>\r\n<h2>Solutions to Odd-Numbered Exercises<\/h2>\r\n4. Find a common denominator and condense your partial fractions back into a single fraction. If it matches the original expression it was decomposed correctly.\r\n\r\n7.\u00a0[latex]\\frac{8}{x+3}-\\frac{5}{x - 8}[\/latex]\r\n\r\n11.\u00a0[latex]\\frac{3}{5x - 2}+\\frac{4}{4x - 1}[\/latex]\r\n\r\n13.\u00a0[latex]\\frac{5}{2\\left(x+3\\right)}+\\frac{5}{2\\left(x - 3\\right)}[\/latex]\r\n\r\n17.\u00a0[latex]\\frac{9}{5\\left(x+2\\right)}+\\frac{11}{5\\left(x - 3\\right)}[\/latex]\r\n\r\n21.\u00a0[latex]\\frac{1}{x - 2}+\\frac{2}{{\\left(x - 2\\right)}^{2}}[\/latex]\r\n\r\n23.\u00a0[latex]-\\frac{6}{4x+5}+\\frac{3}{{\\left(4x+5\\right)}^{2}}[\/latex]\r\n\r\n27.\u00a0[latex]\\frac{4}{x}-\\frac{3}{2\\left(x+1\\right)}+\\frac{7}{2{\\left(x+1\\right)}^{2}}[\/latex]\r\n\r\n29.\u00a0[latex]\\frac{4}{x}+\\frac{2}{{x}^{2}}-\\frac{3}{3x+2}+\\frac{7}{2{\\left(3x+2\\right)}^{2}}[\/latex]\r\n\r\n31.\u00a0[latex]\\frac{x+1}{{x}^{2}+x+3}+\\frac{3}{x+2}[\/latex]\r\n\r\n33.\u00a0[latex]\\frac{4 - 3x}{{x}^{2}+3x+8}+\\frac{1}{x - 1}[\/latex]\r\n\r\n37.\u00a0[latex]\\frac{1}{{x}^{2}+x+1}+\\frac{4}{x - 1}[\/latex]\r\n\r\n41.\u00a0[latex]-\\frac{1}{4{x}^{2}+6x+9}+\\frac{1}{2x - 3}[\/latex]\r\n\r\n43.\u00a0[latex]\\frac{1}{x}+\\frac{1}{x+6}-\\frac{4x}{{x}^{2}-6x+36}[\/latex]\r\n\r\n45.\u00a0[latex]\\frac{x+6}{{x}^{2}+1}+\\frac{4x+3}{{\\left({x}^{2}+1\\right)}^{2}}[\/latex]\r\n\r\n47.\u00a0[latex]\\frac{x+1}{x+2}+\\frac{2x+3}{{\\left(x+2\\right)}^{2}}[\/latex]\r\n\r\n49.\u00a0[latex]\\frac{1}{{x}^{2}+3x+25}-\\frac{3x}{{\\left({x}^{2}+3x+25\\right)}^{2}}[\/latex]\r\n\r\n51.\u00a0[latex]\\frac{1}{8x}-\\frac{x}{8\\left({x}^{2}+4\\right)}+\\frac{10-x}{2{\\left({x}^{2}+4\\right)}^{2}}[\/latex]\r\n\r\n53.\u00a0[latex]-\\frac{16}{x}-\\frac{9}{{x}^{2}}+\\frac{16}{x - 1}-\\frac{7}{{\\left(x - 1\\right)}^{2}}[\/latex]\r\n\r\n57.\u00a0[latex]\\frac{5}{x - 2}-\\frac{3}{10\\left(x+2\\right)}+\\frac{7}{x+8}-\\frac{7}{10\\left(x - 8\\right)}[\/latex]\r\n\r\n59.\u00a0[latex]-\\frac{5}{4x}-\\frac{5}{2\\left(x+2\\right)}+\\frac{11}{2\\left(x+4\\right)}+\\frac{5}{4\\left(x+4\\right)}[\/latex]","rendered":"<h2>Systems of Linear Equations: Two Variables<\/h2>\n<p>1.\u00a0No, you can either have zero, one, or infinitely many. Examine graphs.<\/p>\n<p>5.\u00a0You can solve by substitution (isolating [latex]x[\/latex] or [latex]y[\/latex] ), graphically, or by addition.<\/p>\n<p>7.\u00a0Yes<\/p>\n<p>9.\u00a0Yes<\/p>\n<p>11.\u00a0[latex]\\left(-1,2\\right)[\/latex]<\/p>\n<p>13.\u00a0[latex]\\left(-3,1\\right)[\/latex]<\/p>\n<p>15.\u00a0[latex]\\left(-\\frac{3}{5},0\\right)[\/latex]<\/p>\n<p>17.\u00a0No solutions exist.<\/p>\n<p>19.\u00a0[latex]\\left(\\frac{72}{5},\\frac{132}{5}\\right)[\/latex]<\/p>\n<p>21.\u00a0[latex]\\left(6,-6\\right)[\/latex]<\/p>\n<p>23.\u00a0[latex]\\left(-\\frac{1}{2},\\frac{1}{10}\\right)[\/latex]<\/p>\n<p>25.\u00a0No solutions exist.<\/p>\n<p>27.\u00a0[latex]\\left(-\\frac{1}{5},\\frac{2}{3}\\right)[\/latex]<\/p>\n<p>29.\u00a0[latex]\\left(x,\\frac{x+3}{2}\\right)[\/latex]<\/p>\n<p>31.\u00a0[latex]\\left(-4,4\\right)[\/latex]<\/p>\n<p>33.\u00a0[latex]\\left(\\frac{1}{2},\\frac{1}{8}\\right)[\/latex]<\/p>\n<p>35.\u00a0[latex]\\left(\\frac{1}{6},0\\right)[\/latex]<\/p>\n<p>37.\u00a0[latex]\\left(x,2\\left(7x - 6\\right)\\right)[\/latex]<\/p>\n<p>39.\u00a0[latex]\\left(-\\frac{5}{6},\\frac{4}{3}\\right)[\/latex]<\/p>\n<p>41.\u00a0Consistent with one solution<\/p>\n<p>43.\u00a0Consistent with one solution<\/p>\n<p>45.\u00a0Dependent with infinitely many solutions<\/p>\n<p>47.\u00a0[latex]\\left(-3.08,4.91\\right)[\/latex]<\/p>\n<p>49.\u00a0[latex]\\left(-1.52,2.29\\right)[\/latex]<\/p>\n<p>61.\u00a0The numbers are 7.5 and 20.5.<\/p>\n<p>63.\u00a024,000<\/p>\n<p>65.\u00a0790 sophomores, 805 freshman<\/p>\n<p>67.\u00a056 men, 74 women<\/p>\n<p>69.\u00a010 gallons of 10% solution, 15 gallons of 60% solution<\/p>\n<p>73.\u00a0$12,500 in the first account, $10,500 in the second account.<\/p>\n<h2>Systems of Linear Equations: Three Variables<\/h2>\n<p>1.\u00a0No, there can be only one, zero, or infinitely many solutions.<\/p>\n<p>7. No<\/p>\n<p>9.\u00a0Yes<\/p>\n<p>11.\u00a0[latex]\\left(-1,4,2\\right)[\/latex]<\/p>\n<p>13.\u00a0[latex]\\left(-\\frac{85}{107},\\frac{312}{107},\\frac{191}{107}\\right)[\/latex]<\/p>\n<p>15.\u00a0[latex]\\left(1,\\frac{1}{2},0\\right)[\/latex]<\/p>\n<p>17.\u00a0[latex]\\left(4,-6,1\\right)[\/latex]<\/p>\n<p>19.\u00a0[latex]\\left(x,\\frac{1}{27}\\left(65 - 16x\\right),\\frac{x+28}{27}\\right)[\/latex]<\/p>\n<p>21.\u00a0[latex]\\left(-\\frac{45}{13},\\frac{17}{13},-2\\right)[\/latex]<\/p>\n<p>23.\u00a0No solutions exist<\/p>\n<p>25.\u00a0[latex]\\left(0,0,0\\right)[\/latex]<\/p>\n<p>27.\u00a0[latex]\\left(\\frac{4}{7},-\\frac{1}{7},-\\frac{3}{7}\\right)[\/latex]<\/p>\n<p>29.\u00a0[latex]\\left(7,20,16\\right)[\/latex]<\/p>\n<p>39.\u00a0[latex]\\left(\\frac{1}{2},\\frac{1}{5},\\frac{4}{5}\\right)[\/latex]<\/p>\n<p>43.\u00a0[latex]\\left(2,0,0\\right)[\/latex]<\/p>\n<p>51.\u00a024, 36, 48<\/p>\n<p>53.\u00a070 grandparents, 140 parents, 190 children<\/p>\n<p>55.\u00a0Your share was $19.95, Sarah\u2019s share was $40, and your other roommate\u2019s share was $22.05.<\/p>\n<p>57.\u00a0There are infinitely many solutions; we need more information<\/p>\n<p>59.\u00a0500 students, 225 children, and 450 adults<\/p>\n<p>63.\u00a0$400,000 in the account that pays 3% interest, $500,000 in the account that pays 4% interest, and $100,000 in the account that pays 2% interest.<\/p>\n<p>65.\u00a0The United States consumed 26.3%, Japan 7.1%, and China 6.4% of the world\u2019s oil.<\/p>\n<p>69.\u00a0Birds were 19.3%, fish were 18.6%, and mammals were 17.1% of endangered species<\/p>\n<h2>Systems of Nonlinear Equations and Inequalities: Two Variables<\/h2>\n<h2>Solutions to Odd-Numbered Exercises<\/h2>\n<p>3.\u00a0No. There does not need to be a feasible region. Consider a system that is bounded by two parallel lines. One inequality represents the region above the upper line; the other represents the region below the lower line. In this case, no points in the plane are located in both regions; hence there is no feasible region.<\/p>\n<p>7.\u00a0[latex]\\left(0,-3\\right),\\left(3,0\\right)[\/latex]<\/p>\n<p>9.\u00a0[latex]\\left(-\\frac{3\\sqrt{2}}{2},\\frac{3\\sqrt{2}}{2}\\right),\\left(\\frac{3\\sqrt{2}}{2},-\\frac{3\\sqrt{2}}{2}\\right)[\/latex]<\/p>\n<p>11.\u00a0[latex]\\left(-3,0\\right),\\left(3,0\\right)[\/latex]<\/p>\n<p>13.\u00a0[latex]\\left(\\frac{1}{4},-\\frac{\\sqrt{62}}{8}\\right),\\left(\\frac{1}{4},\\frac{\\sqrt{62}}{8}\\right)[\/latex]<\/p>\n<p>15.\u00a0[latex]\\left(-\\frac{\\sqrt{398}}{4},\\frac{199}{4}\\right),\\left(\\frac{\\sqrt{398}}{4},\\frac{199}{4}\\right)[\/latex]<\/p>\n<p>17.\u00a0[latex]\\left(0,2\\right),\\left(1,3\\right)[\/latex]<\/p>\n<p>19.\u00a0[latex]\\left(-\\sqrt{\\frac{1}{2}\\left(\\sqrt{5}-1\\right)},\\frac{1}{2}\\left(1-\\sqrt{5}\\right)\\right),\\left(\\sqrt{\\frac{1}{2}\\left(\\sqrt{5}-1\\right)},\\frac{1}{2}\\left(1-\\sqrt{5}\\right)\\right)[\/latex]<\/p>\n<p>21.\u00a0[latex]\\left(5,0\\right)[\/latex]<\/p>\n<p>25.\u00a0[latex]\\left(3,0\\right)[\/latex]<\/p>\n<p>27.\u00a0No Solutions Exist<\/p>\n<p>33.\u00a0[latex]\\left(2,0\\right)[\/latex]<\/p>\n<p>35.\u00a0[latex]\\left(-\\sqrt{7},-3\\right),\\left(-\\sqrt{7},3\\right),\\left(\\sqrt{7},-3\\right),\\left(\\sqrt{7},3\\right)[\/latex]<\/p>\n<p>41.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181409\/CNX_Precalc_Figure_09_03_2042.jpg\" alt=\"A shaded figure with a dotted line that has two marked points. The first point is at square root of two minus 1, two times (the square root of two minus one). The second point is at negative one minus square root of two, negative two times (one plus the square root of two).\" width=\"487\" height=\"317\" \/><\/p>\n<p>43.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181412\/CNX_Precalc_Figure_09_03_2072.jpg\" alt=\"Two dotted, shaded figures with points marked. The first point is (negative square root of 37 over 2, 3 times square root of seven over two). The second point is (square root of 37 over 2, 3 times square root of 7 over two). The third point is (negative square root 37 over 2, negative 3 times square root 7 divided by 2). The fourth point is (square root 37 over 2, negative 3 times square root of 7 over two).\" width=\"487\" height=\"379\" \/><\/p>\n<p>45.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181415\/CNX_Precalc_Figure_09_03_2092.jpg\" alt=\"Two dotted, shaded figures with marked points. The first point is negative square root of nineteen-tenths, square root of forty-seven-tenths. The second point is square root of 19 tenths, square root of 47 tenths. The third point is negative square root of 19 tenths, negative square root of 47 tenths. The fourth point is square root of 19 tenths, negative square root of 47 tenths.\" width=\"487\" height=\"380\" \/><\/p>\n<p>49.\u00a0[latex]\\left(-2\\sqrt{\\frac{70}{383}},-2\\sqrt{\\frac{35}{29}}\\right),\\left(-2\\sqrt{\\frac{70}{383}},2\\sqrt{\\frac{35}{29}}\\right),\\left(2\\sqrt{\\frac{70}{383}},-2\\sqrt{\\frac{35}{29}}\\right),\\left(2\\sqrt{\\frac{70}{383}},2\\sqrt{\\frac{35}{29}}\\right)[\/latex]<\/p>\n<p>51.\u00a0No Solution Exists<\/p>\n<p>55.\u00a012, 288<\/p>\n<p>57.\u00a02\u201320 computers<\/p>\n<h2>Partial Fractions<\/h2>\n<h2>Solutions to Odd-Numbered Exercises<\/h2>\n<p>4. Find a common denominator and condense your partial fractions back into a single fraction. If it matches the original expression it was decomposed correctly.<\/p>\n<p>7.\u00a0[latex]\\frac{8}{x+3}-\\frac{5}{x - 8}[\/latex]<\/p>\n<p>11.\u00a0[latex]\\frac{3}{5x - 2}+\\frac{4}{4x - 1}[\/latex]<\/p>\n<p>13.\u00a0[latex]\\frac{5}{2\\left(x+3\\right)}+\\frac{5}{2\\left(x - 3\\right)}[\/latex]<\/p>\n<p>17.\u00a0[latex]\\frac{9}{5\\left(x+2\\right)}+\\frac{11}{5\\left(x - 3\\right)}[\/latex]<\/p>\n<p>21.\u00a0[latex]\\frac{1}{x - 2}+\\frac{2}{{\\left(x - 2\\right)}^{2}}[\/latex]<\/p>\n<p>23.\u00a0[latex]-\\frac{6}{4x+5}+\\frac{3}{{\\left(4x+5\\right)}^{2}}[\/latex]<\/p>\n<p>27.\u00a0[latex]\\frac{4}{x}-\\frac{3}{2\\left(x+1\\right)}+\\frac{7}{2{\\left(x+1\\right)}^{2}}[\/latex]<\/p>\n<p>29.\u00a0[latex]\\frac{4}{x}+\\frac{2}{{x}^{2}}-\\frac{3}{3x+2}+\\frac{7}{2{\\left(3x+2\\right)}^{2}}[\/latex]<\/p>\n<p>31.\u00a0[latex]\\frac{x+1}{{x}^{2}+x+3}+\\frac{3}{x+2}[\/latex]<\/p>\n<p>33.\u00a0[latex]\\frac{4 - 3x}{{x}^{2}+3x+8}+\\frac{1}{x - 1}[\/latex]<\/p>\n<p>37.\u00a0[latex]\\frac{1}{{x}^{2}+x+1}+\\frac{4}{x - 1}[\/latex]<\/p>\n<p>41.\u00a0[latex]-\\frac{1}{4{x}^{2}+6x+9}+\\frac{1}{2x - 3}[\/latex]<\/p>\n<p>43.\u00a0[latex]\\frac{1}{x}+\\frac{1}{x+6}-\\frac{4x}{{x}^{2}-6x+36}[\/latex]<\/p>\n<p>45.\u00a0[latex]\\frac{x+6}{{x}^{2}+1}+\\frac{4x+3}{{\\left({x}^{2}+1\\right)}^{2}}[\/latex]<\/p>\n<p>47.\u00a0[latex]\\frac{x+1}{x+2}+\\frac{2x+3}{{\\left(x+2\\right)}^{2}}[\/latex]<\/p>\n<p>49.\u00a0[latex]\\frac{1}{{x}^{2}+3x+25}-\\frac{3x}{{\\left({x}^{2}+3x+25\\right)}^{2}}[\/latex]<\/p>\n<p>51.\u00a0[latex]\\frac{1}{8x}-\\frac{x}{8\\left({x}^{2}+4\\right)}+\\frac{10-x}{2{\\left({x}^{2}+4\\right)}^{2}}[\/latex]<\/p>\n<p>53.\u00a0[latex]-\\frac{16}{x}-\\frac{9}{{x}^{2}}+\\frac{16}{x - 1}-\\frac{7}{{\\left(x - 1\\right)}^{2}}[\/latex]<\/p>\n<p>57.\u00a0[latex]\\frac{5}{x - 2}-\\frac{3}{10\\left(x+2\\right)}+\\frac{7}{x+8}-\\frac{7}{10\\left(x - 8\\right)}[\/latex]<\/p>\n<p>59.\u00a0[latex]-\\frac{5}{4x}-\\frac{5}{2\\left(x+2\\right)}+\\frac{11}{2\\left(x+4\\right)}+\\frac{5}{4\\left(x+4\\right)}[\/latex]<\/p>\n","protected":false},"author":13,"menu_order":9,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":224,"module-header":"- Select Header -","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\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/320"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/320\/revisions"}],"predecessor-version":[{"id":337,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/320\/revisions\/337"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/224"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/320\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=320"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=320"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=320"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=320"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}