{"id":1950,"date":"2021-12-02T19:40:08","date_gmt":"2021-12-03T00:40:08","guid":{"rendered":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/answer-key-8-7\/"},"modified":"2022-11-02T10:38:39","modified_gmt":"2022-11-02T14:38:39","slug":"answer-key-8-7","status":"publish","type":"back-matter","link":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/answer-key-8-7\/","title":{"raw":"Answer Key 8.7","rendered":"Answer Key 8.7"},"content":{"raw":"
    \n \t
  1. [latex]\\text{LCD}=2(x)[\/latex]\n[latex]\\begin{array}[t]{rrcrrrl}\n3x(2x)&-&x&-&2&=&0 \\\\\n6x^2&-&x&-&2&=&0 \\\\\n(3x&-&2)(2x&+&1)&=&0 \\\\ \\\\\n&&&&x&=&\\dfrac{2}{3}, -\\dfrac{1}{2}\n\\end{array}[\/latex]<\/li>\n \t
  2. [latex]\\text{LCD}=x+1[\/latex]\n[latex]\\begin{array}[t]{rrcrrrl}\n(x&+&1)(x&+&1)&=&\\phantom{-}4 \\\\\nx^2&+&2x&+&1&=&\\phantom{-}4 \\\\\n&&&-&4&&-4 \\\\\n\\hline\nx^2&+&2x&-&3&=&0 \\\\\n(x&-&1)(x&+&3)&=&0 \\\\ \\\\\n&&&&x&=&1, -3\n\\end{array}[\/latex]<\/li>\n \t
  3. [latex]\\text{LCD}=x-4[\/latex]\n[latex]\\begin{array}[t]{rrcrrrllrrr}\nx(x&-&4)&+&20&=&5x&-&2(x&-&4) \\\\\nx^2&-&4x&+&20&=&5x&-&2x&+&8 \\\\\n&-&3x&-&8&&&-&3x&-&8 \\\\\n\\hline\nx^2&-&7x&+&12&=&0&&&& \\\\\n(x&-&4)(x&-&3)&=&0&&&& \\\\ \\\\\n&&&&x&=&3,&4&&&\n\\end{array}[\/latex]<\/li>\n \t
  4. [latex]\\text{LCD}=x-1[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrllr}\nx^2&+&6&+&x&-&2&=&\\phantom{-}2x(x&-&1) \\\\\n&&x^2&+&x&+&4&=&\\phantom{-}2x^2&-&2x \\\\\n&-&2x^2&+&2x&&&&-2x^2&+&2x \\\\\n\\hline\n&&-x^2&+&3x&+&4&=&0&& \\\\\n&&x^2&-&3x&-&4&=&0&& \\\\\n&&(x&-&4)(x&+&1)&=&0&& \\\\ \\\\\n&&&&&&x&=&4, 1&& \\\\\n\\end{array}[\/latex]<\/li>\n \t
  5. [latex]\\text{LCD}=x-3[\/latex]\n[latex]\\begin{array}[t]{rrcrrrr}\nx(x&-&3)&+&6&=&2x \\\\\nx^2&-&3x&+&6&=&2x \\\\\n&-&2x&&&&-2x \\\\\n\\hline\nx^2&-&5x&+&6&=&0 \\\\\n(x&-&3)(x&-&2)&=&0 \\\\ \\\\\n&&&&x&=&2, 3\n\\end{array}[\/latex]<\/li>\n \t
  6. [latex]\\text{LCD}=(x-1)(3-x)[\/latex]\n[latex]\\begin{array}[t]{rrcrrrlrrrrrcrr}\n(x&-&4)(3&-&x)&=&\\phantom{-}12(x&-&1)&+&(x&-&1)(3&-&x) \\\\\n-x^2&+&7x&-&12&=&\\phantom{-}12x&-&12&-&x^2&+&4x&-&3 \\\\\n+x^2&-&16x&+&15&&-12x&+&12&+&x^2&-&4x&+&3 \\\\\n\\hline\n&&-9x&+&3&=&0&&&&&&&& \\\\\n&&&&3&=&9x&&&&&&&& \\\\ \\\\\n&&&&x&=&\\dfrac{3}{9}\\hspace{0.1in}\\text{ or}&\\dfrac{1}{3}&&&&&&&\n\\end{array}[\/latex]<\/li>\n \t
  7. [latex]\\text{LCD}=(2m-5)(3m+1)(2)[\/latex]\n[latex]\\begin{array}[t]{rrcrcrrrrrcrr}\n3m(3m&+&1)(2)&-&7(2m&-&5)(2)&=&3(2m&-&5)(3m&+&1) \\\\\n18m^2&+&6m&-&28m&+&70&=&18m^2&-&39m&-&15 \\\\\n-18m^2&&&+&39m&+&15&&-18m^2&+&39m&+&15 \\\\\n\\hline\n&&&&17m&+&85&=&0&&&& \\\\\n&&&&&-&85&&-85&&&& \\\\\n\\hline\n&&&&&&\\dfrac{17m}{17}&=&\\dfrac{-85}{17}&&&& \\\\ \\\\\n&&&&&&m&=&-5&&&&\n\\end{array}[\/latex]<\/li>\n \t
  8. [latex]\\text{LCD}=(1-x)(3-x)[\/latex]\n[latex]\\begin{array}[t]{rrcrrrrrr}\n(4&-&x)(3&-&x)&=&12(1&-&x) \\\\\n12&-&7x&+&x^2&=&12&-&12x \\\\\n-12&+&12x&&&&-12&+&12x \\\\\n\\hline\n&&x^2&+&5x&=&0&& \\\\\n&&x(x&+&5)&=&0&& \\\\ \\\\\n&&&&x&=&0,&-5&\n\\end{array}[\/latex]<\/li>\n \t
  9. [latex]\\text{LCD}=2(y-3)(y-4)[\/latex]\n[latex]\\begin{array}[t]{crrrrrcrrrrrcrr}\n7(2)(y&-&4)&-&1(y&-&3)(y&-&4)&=&(y&-&2)(2)(y&-&3) \\\\\n14y&-&56&-&y^2&+&7y&-&12&=&2y^2&-&10y&+&12 \\\\ \\\\\n&&&&-\\phantom{0}y^2&+&21y&-&68&=&2y^2&-&10y&+&12 \\\\\n&&&&-2y^2&+&10y&-&12&&-2y^2&+&10y&-&12 \\\\\n\\hline\n&&&&-3y^2&+&31y&-&80&=&0&&&& \\\\\n&&&&3y^2&-&31y&+&80&=&0&&&& \\\\\n&&&&(y&-&5)(3y&-&16)&=&0&&&& \\\\ \\\\\n&&&&&&&&y&=&5, &\\dfrac{16}{3}&&&\n\\end{array}[\/latex]<\/li>\n \t
  10. [latex]\\text{LCD}=(x+2)(x-2)[\/latex]\n[latex]\\begin{array}[t]{rrrrrrrrrrr}\n1(x&-&2)&+&1(x&+&2)&=&3x&+&8 \\\\\nx&-&2&+&x&+&2&=&3x&+&8 \\\\\n&&&&-2x&&&&-2x&& \\\\\n\\hline\n&&&&&&0&=&x&+&8 \\\\\n&&&&&&-8&&&-&8 \\\\\n\\hline\n&&&&&&x&=&-8&&\n\\end{array}[\/latex]<\/li>\n \t
  11. [latex]\\text{LCD}=(x+1)(x-1)(6)[\/latex]\n[latex]\\begin{array}[t]{rrcrcrrrcrcrrrcrr}\n(x&+&1)(x&+&1)(6)&-&(x&-&1)(x&-&1)(6)&=&5(x&+&1)(x&-&1) \\\\\n6(x^2&+&2x&+&1)&-&6(x^2&-&2x&+&1)&=&5(x^2&&-&&1) \\\\\n6x^2&+&12x&+&6&-&6x^2&+&12x&-&6&=&5x^2&&&-&5 \\\\\n&&&&&&&&&&24x&=&5x^2&&&-&5 \\\\\n&&&&&&&&&&-24x&&&-&24x&& \\\\\n\\hline\n&&&&&&&&&&0&=&5x^2&-&24x&-&5 \\\\\n&&&&&&&&&&0&=&(5x&+&1)(x&-&5) \\\\ \\\\\n&&&&&&&&&&x&=&5, &-\\dfrac{1}{5}&&&\n\\end{array}[\/latex]<\/li>\n \t
  12. [latex]\\text{LCD}=(x+3)(x-2)[\/latex]\n[latex]\\begin{array}[t]{rrcrcrrrrrr}\n(x&-&2)(x&-&2)&-&1(x&+&3)&=&1 \\\\\nx^2&-&4x&+&4&-&x&-&3&=&1 \\\\\n&&&&&&&-&1&&-1 \\\\\n\\hline\n&&&&&&x^2&-&5x&=&0 \\\\\n&&&&&&x(x&-&5)&=&0 \\\\ \\\\\n&&&&&&&&x&=&0, 5\n\\end{array}[\/latex]<\/li>\n \t
  13. [latex]\\text{LCD}=(x-1)(x+1)[\/latex]\n[latex]\\begin{array}[t]{rrrrcrrrrrcrr}\nx(x&+&1)&-&2(x&-&1)&=&4x^2&&&& \\\\\nx^2&+&x&-&2x&+&2&=&4x^2&&&& \\\\\n-x^2&&&+&x&-&2&&-x^2&+&x&-&2 \\\\\n\\hline\n&&&&&&0&=&3x^2&+&x&-&2 \\\\\n&&&&&&0&=&(3x&-&2)(x&+&1) \\\\ \\\\\n&&&&&&0&=&\\dfrac{2}{3},&-1&&&\n\\end{array}[\/latex]<\/li>\n \t
  14. [latex]\\text{LCD}=(x+2)(x-4)[\/latex]\n[latex]\\begin{array}[t]{rrrrcrrrr}\n2x(x&-&4)&+&2(x&+&2)&=&3x \\\\\n2x^2&-&8x&+&2x&+&4&=&3x \\\\\n&&&-&3x&&&&-3x \\\\\n\\hline\n&&2x^2&-&9x&+&4&=&0 \\\\\n&&(2x&-&1)(x&-&4)&=&0 \\\\ \\\\\n&&&&&&x&=&\\dfrac{1}{2}, 4\n\\end{array}[\/latex]<\/li>\n \t
  15. [latex]\\text{LCD}=(x+1)(x+5)[\/latex]\n[latex]\\begin{array}[t]{rrrrcrrrl}\n2x(x&+&5)&-&3(x&+&1)&=&-8x^2 \\\\\n2x^2&+&10x&-&3x&-&3&=&-8x^2 \\\\\n+8x^2&&&&&&&&+8x^2 \\\\\n\\hline\n&&10x^2&+&7x&-&3&=&0 \\\\\n&&(10x&-&3)(x&+&1)&=&0 \\\\ \\\\\n&&&&&&x&=&\\dfrac{3}{10}, -1\n\\end{array}[\/latex]<\/li>\n<\/ol>","rendered":"
      \n
    1. [latex]\\text{LCD}=2(x)[\/latex]
      \n[latex]\\begin{array}[t]{rrcrrrl} 3x(2x)&-&x&-&2&=&0 \\\\ 6x^2&-&x&-&2&=&0 \\\\ (3x&-&2)(2x&+&1)&=&0 \\\\ \\\\ &&&&x&=&\\dfrac{2}{3}, -\\dfrac{1}{2} \\end{array}[\/latex]<\/li>\n
    2. [latex]\\text{LCD}=x+1[\/latex]
      \n[latex]\\begin{array}[t]{rrcrrrl} (x&+&1)(x&+&1)&=&\\phantom{-}4 \\\\ x^2&+&2x&+&1&=&\\phantom{-}4 \\\\ &&&-&4&&-4 \\\\ \\hline x^2&+&2x&-&3&=&0 \\\\ (x&-&1)(x&+&3)&=&0 \\\\ \\\\ &&&&x&=&1, -3 \\end{array}[\/latex]<\/li>\n
    3. [latex]\\text{LCD}=x-4[\/latex]
      \n[latex]\\begin{array}[t]{rrcrrrllrrr} x(x&-&4)&+&20&=&5x&-&2(x&-&4) \\\\ x^2&-&4x&+&20&=&5x&-&2x&+&8 \\\\ &-&3x&-&8&&&-&3x&-&8 \\\\ \\hline x^2&-&7x&+&12&=&0&&&& \\\\ (x&-&4)(x&-&3)&=&0&&&& \\\\ \\\\ &&&&x&=&3,&4&&& \\end{array}[\/latex]<\/li>\n
    4. [latex]\\text{LCD}=x-1[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrrllr} x^2&+&6&+&x&-&2&=&\\phantom{-}2x(x&-&1) \\\\ &&x^2&+&x&+&4&=&\\phantom{-}2x^2&-&2x \\\\ &-&2x^2&+&2x&&&&-2x^2&+&2x \\\\ \\hline &&-x^2&+&3x&+&4&=&0&& \\\\ &&x^2&-&3x&-&4&=&0&& \\\\ &&(x&-&4)(x&+&1)&=&0&& \\\\ \\\\ &&&&&&x&=&4, 1&& \\\\ \\end{array}[\/latex]<\/li>\n
    5. [latex]\\text{LCD}=x-3[\/latex]
      \n[latex]\\begin{array}[t]{rrcrrrr} x(x&-&3)&+&6&=&2x \\\\ x^2&-&3x&+&6&=&2x \\\\ &-&2x&&&&-2x \\\\ \\hline x^2&-&5x&+&6&=&0 \\\\ (x&-&3)(x&-&2)&=&0 \\\\ \\\\ &&&&x&=&2, 3 \\end{array}[\/latex]<\/li>\n
    6. [latex]\\text{LCD}=(x-1)(3-x)[\/latex]
      \n[latex]\\begin{array}[t]{rrcrrrlrrrrrcrr} (x&-&4)(3&-&x)&=&\\phantom{-}12(x&-&1)&+&(x&-&1)(3&-&x) \\\\ -x^2&+&7x&-&12&=&\\phantom{-}12x&-&12&-&x^2&+&4x&-&3 \\\\ +x^2&-&16x&+&15&&-12x&+&12&+&x^2&-&4x&+&3 \\\\ \\hline &&-9x&+&3&=&0&&&&&&&& \\\\ &&&&3&=&9x&&&&&&&& \\\\ \\\\ &&&&x&=&\\dfrac{3}{9}\\hspace{0.1in}\\text{ or}&\\dfrac{1}{3}&&&&&&& \\end{array}[\/latex]<\/li>\n
    7. [latex]\\text{LCD}=(2m-5)(3m+1)(2)[\/latex]
      \n[latex]\\begin{array}[t]{rrcrcrrrrrcrr} 3m(3m&+&1)(2)&-&7(2m&-&5)(2)&=&3(2m&-&5)(3m&+&1) \\\\ 18m^2&+&6m&-&28m&+&70&=&18m^2&-&39m&-&15 \\\\ -18m^2&&&+&39m&+&15&&-18m^2&+&39m&+&15 \\\\ \\hline &&&&17m&+&85&=&0&&&& \\\\ &&&&&-&85&&-85&&&& \\\\ \\hline &&&&&&\\dfrac{17m}{17}&=&\\dfrac{-85}{17}&&&& \\\\ \\\\ &&&&&&m&=&-5&&&& \\end{array}[\/latex]<\/li>\n
    8. [latex]\\text{LCD}=(1-x)(3-x)[\/latex]
      \n[latex]\\begin{array}[t]{rrcrrrrrr} (4&-&x)(3&-&x)&=&12(1&-&x) \\\\ 12&-&7x&+&x^2&=&12&-&12x \\\\ -12&+&12x&&&&-12&+&12x \\\\ \\hline &&x^2&+&5x&=&0&& \\\\ &&x(x&+&5)&=&0&& \\\\ \\\\ &&&&x&=&0,&-5& \\end{array}[\/latex]<\/li>\n
    9. [latex]\\text{LCD}=2(y-3)(y-4)[\/latex]
      \n[latex]\\begin{array}[t]{crrrrrcrrrrrcrr} 7(2)(y&-&4)&-&1(y&-&3)(y&-&4)&=&(y&-&2)(2)(y&-&3) \\\\ 14y&-&56&-&y^2&+&7y&-&12&=&2y^2&-&10y&+&12 \\\\ \\\\ &&&&-\\phantom{0}y^2&+&21y&-&68&=&2y^2&-&10y&+&12 \\\\ &&&&-2y^2&+&10y&-&12&&-2y^2&+&10y&-&12 \\\\ \\hline &&&&-3y^2&+&31y&-&80&=&0&&&& \\\\ &&&&3y^2&-&31y&+&80&=&0&&&& \\\\ &&&&(y&-&5)(3y&-&16)&=&0&&&& \\\\ \\\\ &&&&&&&&y&=&5, &\\dfrac{16}{3}&&& \\end{array}[\/latex]<\/li>\n
    10. [latex]\\text{LCD}=(x+2)(x-2)[\/latex]
      \n[latex]\\begin{array}[t]{rrrrrrrrrrr} 1(x&-&2)&+&1(x&+&2)&=&3x&+&8 \\\\ x&-&2&+&x&+&2&=&3x&+&8 \\\\ &&&&-2x&&&&-2x&& \\\\ \\hline &&&&&&0&=&x&+&8 \\\\ &&&&&&-8&&&-&8 \\\\ \\hline &&&&&&x&=&-8&& \\end{array}[\/latex]<\/li>\n
    11. [latex]\\text{LCD}=(x+1)(x-1)(6)[\/latex]
      \n[latex]\\begin{array}[t]{rrcrcrrrcrcrrrcrr} (x&+&1)(x&+&1)(6)&-&(x&-&1)(x&-&1)(6)&=&5(x&+&1)(x&-&1) \\\\ 6(x^2&+&2x&+&1)&-&6(x^2&-&2x&+&1)&=&5(x^2&&-&&1) \\\\ 6x^2&+&12x&+&6&-&6x^2&+&12x&-&6&=&5x^2&&&-&5 \\\\ &&&&&&&&&&24x&=&5x^2&&&-&5 \\\\ &&&&&&&&&&-24x&&&-&24x&& \\\\ \\hline &&&&&&&&&&0&=&5x^2&-&24x&-&5 \\\\ &&&&&&&&&&0&=&(5x&+&1)(x&-&5) \\\\ \\\\ &&&&&&&&&&x&=&5, &-\\dfrac{1}{5}&&& \\end{array}[\/latex]<\/li>\n
    12. [latex]\\text{LCD}=(x+3)(x-2)[\/latex]
      \n[latex]\\begin{array}[t]{rrcrcrrrrrr} (x&-&2)(x&-&2)&-&1(x&+&3)&=&1 \\\\ x^2&-&4x&+&4&-&x&-&3&=&1 \\\\ &&&&&&&-&1&&-1 \\\\ \\hline &&&&&&x^2&-&5x&=&0 \\\\ &&&&&&x(x&-&5)&=&0 \\\\ \\\\ &&&&&&&&x&=&0, 5 \\end{array}[\/latex]<\/li>\n
    13. [latex]\\text{LCD}=(x-1)(x+1)[\/latex]
      \n[latex]\\begin{array}[t]{rrrrcrrrrrcrr} x(x&+&1)&-&2(x&-&1)&=&4x^2&&&& \\\\ x^2&+&x&-&2x&+&2&=&4x^2&&&& \\\\ -x^2&&&+&x&-&2&&-x^2&+&x&-&2 \\\\ \\hline &&&&&&0&=&3x^2&+&x&-&2 \\\\ &&&&&&0&=&(3x&-&2)(x&+&1) \\\\ \\\\ &&&&&&0&=&\\dfrac{2}{3},&-1&&& \\end{array}[\/latex]<\/li>\n
    14. [latex]\\text{LCD}=(x+2)(x-4)[\/latex]
      \n[latex]\\begin{array}[t]{rrrrcrrrr} 2x(x&-&4)&+&2(x&+&2)&=&3x \\\\ 2x^2&-&8x&+&2x&+&4&=&3x \\\\ &&&-&3x&&&&-3x \\\\ \\hline &&2x^2&-&9x&+&4&=&0 \\\\ &&(2x&-&1)(x&-&4)&=&0 \\\\ \\\\ &&&&&&x&=&\\dfrac{1}{2}, 4 \\end{array}[\/latex]<\/li>\n
    15. [latex]\\text{LCD}=(x+1)(x+5)[\/latex]
      \n[latex]\\begin{array}[t]{rrrrcrrrl} 2x(x&+&5)&-&3(x&+&1)&=&-8x^2 \\\\ 2x^2&+&10x&-&3x&-&3&=&-8x^2 \\\\ +8x^2&&&&&&&&+8x^2 \\\\ \\hline &&10x^2&+&7x&-&3&=&0 \\\\ &&(10x&-&3)(x&+&1)&=&0 \\\\ \\\\ &&&&&&x&=&\\dfrac{3}{10}, -1 \\end{array}[\/latex]<\/li>\n<\/ol>\n","protected":false},"author":90,"menu_order":80,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"cc-by-nc-sa"},"back-matter-type":[],"contributor":[],"license":[56],"class_list":["post-1950","back-matter","type-back-matter","status-publish","hentry","license-cc-by-nc-sa"],"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1950","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter"}],"about":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/types\/back-matter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/users\/90"}],"version-history":[{"count":1,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1950\/revisions"}],"predecessor-version":[{"id":1951,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1950\/revisions\/1951"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1950\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/media?parent=1950"}],"wp:term":[{"taxonomy":"back-matter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter-type?post=1950"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/contributor?post=1950"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/license?post=1950"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}