{"id":1946,"date":"2021-12-02T19:40:07","date_gmt":"2021-12-03T00:40:07","guid":{"rendered":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/answer-key-8-5\/"},"modified":"2024-10-18T12:36:26","modified_gmt":"2024-10-18T16:36:26","slug":"answer-key-8-5","status":"publish","type":"back-matter","link":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/answer-key-8-5\/","title":{"raw":"Answer Key 8.5","rendered":"Answer Key 8.5"},"content":{"raw":"
    \r\n \t
  1. [latex]\\dfrac{\\left(1+\\dfrac{1}{x}\\right)x^2}{\\left(1-\\dfrac{1}{x^2}\\right)x^2}\\Rightarrow \\dfrac{x^2+x}{x^2-1}\\Rightarrow \\dfrac{x\\cancel{(x+1)}}{\\cancel{(x+1)}(x-1)}\\Rightarrow \\dfrac{x}{x-1}[\/latex]<\/li>\r\n \t
  2. [latex]\\dfrac{\\left(1-\\dfrac{1}{y^2}\\right)y^2}{\\left(1+\\dfrac{1}{y}\\right)y^2}\\Rightarrow \\dfrac{y^2-1}{y^2+y}\\Rightarrow \\dfrac{(y-1)\\cancel{(y+1)}}{y\\cancel{(y+1)}}\\Rightarrow \\dfrac{y-1}{y}[\/latex]<\/li>\r\n \t
  3. [latex]\\dfrac{\\left(\\dfrac{a}{b}+2\\right)b^2}{\\left(\\dfrac{a^2}{b^2}-4\\right)b^2}\\Rightarrow \\dfrac{ab+2b^2}{a^2-4b^2}\\Rightarrow \\dfrac{b\\cancel{(a+2b)}}{\\cancel{(a+2b)}(a-2b)}\\Rightarrow \\dfrac{b}{a-2b}[\/latex]<\/li>\r\n \t
  4. [latex]\\dfrac{\\left(\\dfrac{1}{y^2}-9\\right)y^2}{\\left(\\dfrac{1}{y}+3\\right)y^2}\\Rightarrow \\dfrac{1-9y^2}{y+3y^2}\\Rightarrow \\dfrac{(1-3y)\\cancel{(1+3y)}}{y\\cancel{(1+3y)}}\\Rightarrow \\dfrac{1-3y}{y}[\/latex]<\/li>\r\n \t
  5. [latex]\\dfrac{\\left(\\dfrac{1}{a^2}-\\dfrac{1}{a}\\right)a^2}{\\left(\\dfrac{1}{a^2}+\\dfrac{1}{a}\\right)a^2}\\Rightarrow \\dfrac{1-a}{1+a}[\/latex]<\/li>\r\n \t
  6. [latex]\\dfrac{\\left(\\dfrac{1}{b}+\\dfrac{1}{2}\\right)2b(b^2-1)}{\\left(\\dfrac{4}{b^2-1}\\right)2b(b^2-1)}\\Rightarrow \\dfrac{2b(b^2-1)+b(b^2-1)}{4(2b)}\\Rightarrow \\dfrac{2b^2-2+b^3-b}{8b}\\Rightarrow [\/latex]\r\n[latex]\\phantom{a}[\/latex]\r\n[latex]\\dfrac{b^3+2b^2-b-2}{8b}\\Rightarrow \\dfrac{(b-1)(b+1)(b+2)}{8b}[\/latex]<\/li>\r\n \t
  7. [latex]\\dfrac{\\left(x+2-\\dfrac{9}{x+2}\\right)(x+2)}{\\left(x+1+\\dfrac{x-7}{x+2}\\right)(x+2)}\\Rightarrow \\dfrac{(x+2)(x+2)-9}{(x+1)(x+2)+x-7}\\Rightarrow \\dfrac{x^2+4x+4-9}{x^2+3x+2+x-7}\\Rightarrow[\/latex]\r\n[latex]\\phantom{a}[\/latex]\r\n[latex]\\dfrac{x^2+4x-5}{x^2+4x-5}\\Rightarrow 1[\/latex]<\/li>\r\n \t
  8. [latex]\\dfrac{\\left(a-3+\\dfrac{a-3}{a+2}\\right)(a+2)}{\\left(a+4-\\dfrac{4a-5}{a+2}\\right)(a+2)}\\Rightarrow \\dfrac{(a-3)(a+2)+a-3}{(a+4)(a+2)-4a-5}\\Rightarrow \\dfrac{a^2+2a-3a-6+a-3}{a^2+2a+4a+8-4a-5}[\/latex]\r\n[latex]\\phantom{a}[\/latex]\r\n[latex]\\Rightarrow\\dfrac{a^2-9}{a^2+2a+3}\\Rightarrow \\dfrac{(a-3)(a+3)}{a^2+2a+3}[\/latex]<\/li>\r\n \t
  9. [latex]\\dfrac{\\left(\\dfrac{x+y}{y}+\\dfrac{y}{x-y}\\right)y(x-y)}{\\left(\\dfrac{y}{x-y}\\right)y(x-y)}\\Rightarrow \\dfrac{(x+y)(x-y)+y(y)}{y(y)}\\Rightarrow[\/latex]\r\n[latex]\\phantom{a}[\/latex]\r\n[latex]\\dfrac{x^2-y^2+y^2}{y^2}\\Rightarrow \\dfrac{x^2}{y^2}[\/latex]<\/li>\r\n \t
  10. [latex]\\dfrac{\\left(\\dfrac{a-b}{a}-\\dfrac{a}{a+b}\\right)a(a+b)}{\\left(\\dfrac{b^2}{a+b}\\right)a(a+b)}\\Rightarrow \\dfrac{(a-b)(a+b)-a(a)}{b^2(a)}\\Rightarrow [\/latex]\r\n[latex]\\phantom{a}[\/latex]\r\n[latex]\\dfrac{a^2-b^2-a^2}{ab^2}\\Rightarrow \\dfrac{-b^2}{ab^2}[\/latex]<\/li>\r\n \t
  11. [latex]\\dfrac{\\left(\\dfrac{x-y}{y}+\\dfrac{x+y}{x-y}\\right)y(x-y)}{\\left(\\dfrac{y}{x-y}\\right)y(x-y)}\\Rightarrow \\dfrac{(x-y)(x-y)+(x+y)(y)}{y(y)}\\Rightarrow[\/latex]\r\n[latex]\\phantom{a}[\/latex]\r\n[latex]\\dfrac{x^2-2xy+y^2+xy+y^2}{y^2}\\Rightarrow \\dfrac{x^2-xy+2y^2}{y^2}[\/latex]<\/li>\r\n \t
  12. [latex]\\dfrac{\\left(\\dfrac{x-2}{x+2}-\\dfrac{x+2}{x-2}\\right)(x+2)(x-2)}{\\left(\\dfrac{x-2}{x+2}+\\dfrac{x+2}{x-2}\\right)(x+2)(x-2)}\\Rightarrow \\dfrac{x^2-4x+4-(x^2+4x+4)}{x^2-4x+4+x^2+4x+4}\\Rightarrow \\dfrac{-8x}{2x^2+8}\\Rightarrow [\/latex]\r\n[latex]\\phantom{a}[\/latex]\r\n[latex]\\dfrac{\\cancel{2}(-4x)}{\\cancel{2}(x^2+4)}\\Rightarrow \\dfrac{-4x}{x^2+4}[\/latex]<\/li>\r\n<\/ol>","rendered":"
      \n
    1. [latex]\\dfrac{\\left(1+\\dfrac{1}{x}\\right)x^2}{\\left(1-\\dfrac{1}{x^2}\\right)x^2}\\Rightarrow \\dfrac{x^2+x}{x^2-1}\\Rightarrow \\dfrac{x\\cancel{(x+1)}}{\\cancel{(x+1)}(x-1)}\\Rightarrow \\dfrac{x}{x-1}[\/latex]<\/li>\n
    2. [latex]\\dfrac{\\left(1-\\dfrac{1}{y^2}\\right)y^2}{\\left(1+\\dfrac{1}{y}\\right)y^2}\\Rightarrow \\dfrac{y^2-1}{y^2+y}\\Rightarrow \\dfrac{(y-1)\\cancel{(y+1)}}{y\\cancel{(y+1)}}\\Rightarrow \\dfrac{y-1}{y}[\/latex]<\/li>\n
    3. [latex]\\dfrac{\\left(\\dfrac{a}{b}+2\\right)b^2}{\\left(\\dfrac{a^2}{b^2}-4\\right)b^2}\\Rightarrow \\dfrac{ab+2b^2}{a^2-4b^2}\\Rightarrow \\dfrac{b\\cancel{(a+2b)}}{\\cancel{(a+2b)}(a-2b)}\\Rightarrow \\dfrac{b}{a-2b}[\/latex]<\/li>\n
    4. [latex]\\dfrac{\\left(\\dfrac{1}{y^2}-9\\right)y^2}{\\left(\\dfrac{1}{y}+3\\right)y^2}\\Rightarrow \\dfrac{1-9y^2}{y+3y^2}\\Rightarrow \\dfrac{(1-3y)\\cancel{(1+3y)}}{y\\cancel{(1+3y)}}\\Rightarrow \\dfrac{1-3y}{y}[\/latex]<\/li>\n
    5. [latex]\\dfrac{\\left(\\dfrac{1}{a^2}-\\dfrac{1}{a}\\right)a^2}{\\left(\\dfrac{1}{a^2}+\\dfrac{1}{a}\\right)a^2}\\Rightarrow \\dfrac{1-a}{1+a}[\/latex]<\/li>\n
    6. [latex]\\dfrac{\\left(\\dfrac{1}{b}+\\dfrac{1}{2}\\right)2b(b^2-1)}{\\left(\\dfrac{4}{b^2-1}\\right)2b(b^2-1)}\\Rightarrow \\dfrac{2b(b^2-1)+b(b^2-1)}{4(2b)}\\Rightarrow \\dfrac{2b^2-2+b^3-b}{8b}\\Rightarrow[\/latex]
      \n[latex]\\phantom{a}[\/latex]
      \n[latex]\\dfrac{b^3+2b^2-b-2}{8b}\\Rightarrow \\dfrac{(b-1)(b+1)(b+2)}{8b}[\/latex]<\/li>\n
    7. [latex]\\dfrac{\\left(x+2-\\dfrac{9}{x+2}\\right)(x+2)}{\\left(x+1+\\dfrac{x-7}{x+2}\\right)(x+2)}\\Rightarrow \\dfrac{(x+2)(x+2)-9}{(x+1)(x+2)+x-7}\\Rightarrow \\dfrac{x^2+4x+4-9}{x^2+3x+2+x-7}\\Rightarrow[\/latex]
      \n[latex]\\phantom{a}[\/latex]
      \n[latex]\\dfrac{x^2+4x-5}{x^2+4x-5}\\Rightarrow 1[\/latex]<\/li>\n
    8. [latex]\\dfrac{\\left(a-3+\\dfrac{a-3}{a+2}\\right)(a+2)}{\\left(a+4-\\dfrac{4a-5}{a+2}\\right)(a+2)}\\Rightarrow \\dfrac{(a-3)(a+2)+a-3}{(a+4)(a+2)-4a-5}\\Rightarrow \\dfrac{a^2+2a-3a-6+a-3}{a^2+2a+4a+8-4a-5}[\/latex]
      \n[latex]\\phantom{a}[\/latex]
      \n[latex]\\Rightarrow\\dfrac{a^2-9}{a^2+2a+3}\\Rightarrow \\dfrac{(a-3)(a+3)}{a^2+2a+3}[\/latex]<\/li>\n
    9. [latex]\\dfrac{\\left(\\dfrac{x+y}{y}+\\dfrac{y}{x-y}\\right)y(x-y)}{\\left(\\dfrac{y}{x-y}\\right)y(x-y)}\\Rightarrow \\dfrac{(x+y)(x-y)+y(y)}{y(y)}\\Rightarrow[\/latex]
      \n[latex]\\phantom{a}[\/latex]
      \n[latex]\\dfrac{x^2-y^2+y^2}{y^2}\\Rightarrow \\dfrac{x^2}{y^2}[\/latex]<\/li>\n
    10. [latex]\\dfrac{\\left(\\dfrac{a-b}{a}-\\dfrac{a}{a+b}\\right)a(a+b)}{\\left(\\dfrac{b^2}{a+b}\\right)a(a+b)}\\Rightarrow \\dfrac{(a-b)(a+b)-a(a)}{b^2(a)}\\Rightarrow[\/latex]
      \n[latex]\\phantom{a}[\/latex]
      \n[latex]\\dfrac{a^2-b^2-a^2}{ab^2}\\Rightarrow \\dfrac{-b^2}{ab^2}[\/latex]<\/li>\n
    11. [latex]\\dfrac{\\left(\\dfrac{x-y}{y}+\\dfrac{x+y}{x-y}\\right)y(x-y)}{\\left(\\dfrac{y}{x-y}\\right)y(x-y)}\\Rightarrow \\dfrac{(x-y)(x-y)+(x+y)(y)}{y(y)}\\Rightarrow[\/latex]
      \n[latex]\\phantom{a}[\/latex]
      \n[latex]\\dfrac{x^2-2xy+y^2+xy+y^2}{y^2}\\Rightarrow \\dfrac{x^2-xy+2y^2}{y^2}[\/latex]<\/li>\n
    12. [latex]\\dfrac{\\left(\\dfrac{x-2}{x+2}-\\dfrac{x+2}{x-2}\\right)(x+2)(x-2)}{\\left(\\dfrac{x-2}{x+2}+\\dfrac{x+2}{x-2}\\right)(x+2)(x-2)}\\Rightarrow \\dfrac{x^2-4x+4-(x^2+4x+4)}{x^2-4x+4+x^2+4x+4}\\Rightarrow \\dfrac{-8x}{2x^2+8}\\Rightarrow[\/latex]
      \n[latex]\\phantom{a}[\/latex]
      \n[latex]\\dfrac{\\cancel{2}(-4x)}{\\cancel{2}(x^2+4)}\\Rightarrow \\dfrac{-4x}{x^2+4}[\/latex]<\/li>\n<\/ol>\n","protected":false},"author":90,"menu_order":78,"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-1946","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\/1946","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":3,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1946\/revisions"}],"predecessor-version":[{"id":2283,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1946\/revisions\/2283"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1946\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/media?parent=1946"}],"wp:term":[{"taxonomy":"back-matter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter-type?post=1946"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/contributor?post=1946"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/license?post=1946"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}