{"id":49,"date":"2017-08-23T07:11:55","date_gmt":"2017-08-23T07:11:55","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=49"},"modified":"2017-09-18T14:57:29","modified_gmt":"2017-09-18T14:57:29","slug":"solving-linear-equations-inequalities","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/solving-linear-equations-inequalities\/","title":{"rendered":"Solving Linear Equations & Inequalities"},"content":{"rendered":"
\n
\u2b05 Previous Lesson<\/a>\u00a0Workshop Index<\/a>\u00a0Next Lesson\u00a0\u27a1<\/a><\/div>\n

<\/p>\n

Solving Linear Equations & Inequalities<\/h1>\n

Objective<\/h4>\n

In this lesson, you will study how to solve linear equations and inequalities.<\/p>\n

Previously Covered:<\/h4>\n

A linear equation<\/em><\/strong> is an equation that\u00a0can be written in the form ax\u00a0+ b = <\/em>0, where a <\/em>and b<\/em> are real numbers and a <\/em>\u2260 0. <\/em><\/p>\n

\n

What is an equation?<\/strong><\/h3>\n

An equation<\/abbr> is a statement showing that two mathematical expressions are\u00a0equal. Thus, the following are equations.<\/p>\n

<\/p>\n

An equation in the form ax\u00a0+ b = 0<\/em> is a linear\u00a0equation in one variable<\/abbr>. It is also called a first-degree equation in one\u00a0variable<\/em><\/strong>.<\/p>\n

\n

Important Tidbit<\/h4>\n

If an equation has no solution, it is called a contradiction<\/abbr>.<\/p>\n

<\/p>\n

Equations that are true for all values of x<\/em> are called identities<\/abbr>.<\/p>\n

<\/p>\n

If an equation is true for a finite number of values for x<\/em>, it is called a conditional equation<\/abbr>.<\/p>\n

<\/p>\n

A conditional linear equation has exactly one solution.<\/p>\n<\/div>\n

How do we solve a linear equation?<\/h3>\n

To solve a linear equation, manipulate the terms until x<\/em> is isolated on one side. To do so,\u00a0follow these
\nsteps.<\/p>\n

    \n
  1. Simplify both sides of the equation.<\/strong> Remove parentheses and combine like terms.<\/li>\n
  2. Clear all fractions<\/strong>. Multiply both sides of the equation by the lowest common denominator.<\/li>\n
  3. Isolate the variable on one side of the equation<\/strong>. Apply the properties of equality.<\/li>\n<\/ol>\n
    \n

    Important Tidbit<\/h4>\n

    1. Addition Property of Equality\u00a0If a = b<\/em>, then .<\/p>\n

    2. Subtraction Property of Equality\u00a0If a = b<\/em>, then .<\/p>\n

    3. Multiplication Property of Equality\u00a0If a = b<\/em>, then , when c<\/em> \u2260 0.<\/p>\n

    4. Division Property of Equality\u00a0If a = b<\/em>, , when c<\/em> \u2260 0.<\/p>\n

    The properties of equality are true for all real\u00a0numbers a, b,<\/em> and c.<\/em><\/p>\n<\/div>\n

    \n

    Question<\/h4>\n
    \n

    Which choice shows the solution to <\/em>?<\/p>\n

      \n
    1. 4<\/li>\n
    2. 8<\/li>\n
    3. 16<\/li>\n
    4. 18<\/li>\n<\/ol>\n<\/div>\n

      Reveal Answer <\/a><\/p>\n

      \n

      The correct answer is A. To solve, first apply the subtraction\u00a0property of equality.<\/p>\n

      <\/em><\/p>\n

      Then apply the division property.<\/p>\n

      <\/em><\/p>\n<\/div>\n<\/section>\n

      \n

      Question<\/h4>\n
      \n

      Which choice shows the solution to ?<\/p>\n

        \n
      1. 3<\/li>\n
      2. 5<\/li>\n
      3. 6<\/li>\n
      4. 18<\/li>\n<\/ol>\n<\/div>\n

        Reveal Answer <\/a><\/p>\n

        \n

        The correct answer is B. To solve, first multiply to remove\u00a0the parentheses, then combine like terms and simplify.<\/p>\n

        <\/em><\/p>\n

        Then apply the subtraction property of equality.<\/p>\n

        <\/em><\/p>\n<\/div>\n<\/section>\n

        What if x appears on both sides of the equation?<\/h3>\n

        Use the same procedure and properties as before to isolate x<\/em> on one side of the equation.<\/p>\n

        \n

        Question<\/h4>\n
        \n

        Which choice shows the value\u00a0of x<\/em> in the equation\u00a0<\/em>?<\/p>\n

          \n
        1. 1<\/li>\n
        2. 2<\/li>\n
        3. 3<\/li>\n
        4. 5<\/li>\n<\/ol>\n<\/div>\n

          Reveal Answer <\/a><\/p>\n

          \n

          The correct answer is C. To solve, first remove the parentheses by distributing.<\/p>\n

          <\/em><\/p>\n

          Then apply the subtraction property of equality, with the 3x<\/em> term, and then the 3.<\/p>\n

          <\/em><\/p>\n<\/div>\n<\/section>\n

          What if there is a fractional coefficient?<\/h4>\n

          If there is a fractional coefficient in the problem, multiply\u00a0by the lowest common denominator if possible.<\/p>\n

          \n

          Question<\/h4>\n
          \n

          Which choice shows the value of x in the equation <\/em>?<\/p>\n

            \n
          1. <\/li>\n
          2. <\/li>\n
          3. 2<\/li>\n
          4. 4<\/li>\n<\/ol>\n<\/div>\n

            Reveal Answer <\/a><\/p>\n

            \n

            The correct answer is D. To solve, first multiply both sides\u00a0by the lowest common denominator.<\/p>\n

            <\/em><\/p>\n

            Then apply the addition property of equality.<\/p>\n

            <\/em><\/p>\n

            Finally, apply the subtraction property of equality.<\/p>\n

            <\/em><\/p>\n<\/div>\n<\/section>\n

            How do we solve linear inequalities?<\/strong><\/h3>\n

            Inequalities<\/abbr> are solved in the same way as equations. In inequalities, however,\u00a0pay attention to the sign.\u00a0The symbol ><\/em> means greater than<\/em>.\u00a0The symbol <<\/em> means less than<\/em>.\u00a0The symbol \u2265<\/em> means greater than or equal to<\/em>.\u00a0The symbol \u2264<\/em> means less than or equal to.\u00a0<\/em>Inequalities contain these symbols. The following are properties of inequalities,\u00a0for all real numbers a,\u00a0b, <\/em>and c.<\/em><\/p>\n

              \n
            1. Property of Symmetry\n

              If a < b,\u00a0<\/em>then b > a. <\/em><\/p>\n<\/li>\n

            2. Transitive Property\n

              If a <\u00a0b <\/em>and b < c, <\/em>then\u00a0a < c.<\/em><\/p>\n<\/li>\n

            3. Addition Property\n

              If a <\u00a0<\/em>b, then a\u00a0+ c <\u00a0b + c. <\/em><\/p>\n<\/li>\n

            4. Subtraction Property\n

              If a <\u00a0<\/em>b, then a\u00a0\u2013 c <\u00a0b \u2013 c. <\/em><\/p>\n<\/li>\n

            5. Positive Multiplication Property\n

              If a <b <\/em>and c >\u00a0<\/em>0,\u00a0<\/em>then a<\/em>(c<\/em>) <\u00a0b<\/em>(c<\/em>).<\/em><\/p>\n<\/li>\n

            6. Negative Multiplication Property\n

              If a <\u00a0<\/em>b and c <\u00a0<\/em>0, \u00a0<\/em>then a<\/em>(c<\/em>) >\u00a0b<\/em>(c<\/em>). <\/em><\/p>\n<\/li>\n

            7. Positive Division Property\n

              If a <\u00a0<\/em>b and c >\u00a0<\/em>0,<\/em> then\u00a0. <\/em><\/p>\n<\/li>\n

            8. Negative Division Property\n

              If a <\u00a0<\/em>b and c <\u00a0<\/em>0, <\/em>then\u00a0.<\/em><\/p>\n

              \n

              Important Tidbit<\/h4>\n

              Note that there are two multiplication and two\u00a0division properties. The order of an inequality reverses if both sides are multiplied or\u00a0divided by a negative number.<\/p>\n<\/div>\n

              \n

              Question<\/h4>\n
              \n

              Which choice shows the value of x in 5x<\/em> + 3 > 2(x<\/em> + 3)?<\/p>\n

                \n
              1. x <\/em>> 1<\/li>\n
              2. x <\/em>> 2<\/li>\n
              3. x <\/em>> 3<\/li>\n
              4. x <\/em>> 4<\/li>\n<\/ol>\n<\/div>\n

                Reveal Answer <\/a><\/p>\n

                \n

                The correct answer is A. To solve, first remove parentheses by\u00a0distributing, and then combine like terms.<\/p>\n

                <\/em><\/p>\n

                Then apply the subtraction property of inequality.<\/p>\n

                <\/em><\/p>\n

                Finally, apply the positive division property.<\/p>\n

                <\/em><\/p>\n<\/div>\n<\/section>\n

                The previous example showed an inequality in two parts: the\u00a0left side and the right side. Inequalities can also have three\u00a0parts: a left side, a right side, and a middle.<\/p>\n

                \n

                Question<\/h4>\n
                \n

                Which choice shows the value of x<\/em> in <\/em> ?<\/p>\n

                  \n
                1. <\/li>\n
                2. <\/li>\n
                3. <\/li>\n
                4. <\/li>\n<\/ol>\n<\/div>\n

                  Reveal Answer <\/a><\/p>\n

                  \n

                  The correct answer is B. To solve, first apply the subtraction\u00a0property of inequality.<\/p>\n

                  <\/em><\/p>\n

                  Then apply the positive division property.<\/p>\n

                  <\/em><\/p>\n<\/div>\n<\/section>\n<\/li>\n<\/ol>\n<\/section>\n

                  <\/p>\n

                  \u2b05 Previous Lesson<\/a>\u00a0Workshop Index<\/a>\u00a0Next Lesson \u27a1<\/a><\/div>\n

                  Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

                  \u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson\u00a0\u27a1 Solving Linear Equations & Inequalities Objective In this lesson, you will study how to solve linear equations and inequalities. Previously Covered: A linear equation is an equation that\u00a0can be written in the form ax\u00a0+ b = 0, where a and b are real numbers and a \u2260 0. What is […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-49","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/49","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=49"}],"version-history":[{"count":11,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/49\/revisions"}],"predecessor-version":[{"id":726,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/49\/revisions\/726"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=49"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}