{"id":4665,"date":"2020-04-21T17:23:53","date_gmt":"2020-04-21T17:23:53","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/chapter\/summary-review-6\/"},"modified":"2021-02-05T23:56:30","modified_gmt":"2021-02-05T23:56:30","slug":"summary-review-6","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/summary-review-6\/","title":{"raw":"Summary: Review Topics","rendered":"Summary: Review Topics"},"content":{"raw":"\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>The addition property of equality states, for all real numbers [latex]a, b, \\text{ and } c: \\text{ if } a=b \\text{, then } a+c=b+c[\/latex]. That is, we may add or subtract the same amount to both entire sides of an equation without changing its value.<\/li>\n \t<li>The multiplication property of equality states, for all real numbers [latex]a, b, \\text{ and } c: \\text{ if } a=b \\text{, then } a \\cdot c=b \\cdot c[\/latex]. That is, we may multiply or divide the same amount to both entire sides of an equation without changing its value.<\/li>\n \t<li>Any point graphed in the coordinate plane is of form [latex]\\left(x, y\\right)[\/latex] where [latex]x[\/latex] is called the x-coordinate and [latex]y[\/latex] is called the y-coordinate. With these coordinates, any point in the plane may be unambiguously located or identified.<\/li>\n \t<li>The coordinates of any ordered pair contained in the graph of an equation satisfies the equation (makes it a true statement when substituted for x and y).<\/li>\n \t<li>The slope of a line, of form [latex]m=\\dfrac{\\text{rise}}{\\text{run}}[\/latex], is a measure of the steepness of a line.<\/li>\n \t<li>Parallel lines have identical slopes; perpendicular lines have opposite, reciprocal slopes.<\/li>\n<\/ul>\n<h2>Key Expressions, Equations, and Inequalities<\/h2>\n<ul>\n \t<li>[latex] \\displaystyle \\text{Slope }=\\frac{\\text{rise}}{\\text{run}}[\/latex] and [latex] \\displaystyle m=\\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}[\/latex] where [latex]m=\\text{slope}[\/latex]&nbsp;and [latex] \\displaystyle ({{x}_{1}},{{y}_{1}})[\/latex] and [latex] \\displaystyle ({{x}_{2}},{{y}_{2}})[\/latex] are two points on the line.<\/li>\n \t<li>For any number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>,&nbsp;[latex]\\left(x^{a}\\right)\\left(x^{b}\\right) = x^{a+b}[\/latex].<\/li>\n \t<li>For any non-zero number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>: [latex] \\displaystyle \\frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[\/latex]<\/li>\n \t<li>For any positive number <i>x<\/i> and integers <i>a<\/i> and <i>b<\/i>: [latex]\\left(x^{a}\\right)^{b}=x^{a\\cdot{b}}[\/latex].<\/li>\n \t<li>For any nonzero numbers <i>a<\/i> and <i>b<\/i> and any integer <i>x<\/i>, [latex]\\left(ab\\right)^{x}=a^{x}\\cdot{b^{x}}[\/latex].<\/li>\n \t<li>For any number <i>a<\/i>, any non-zero number <i>b<\/i>, and any integer <i>x<\/i>, [latex] \\displaystyle {\\left(\\frac{a}{b}\\right)}^{x}=\\frac{a^{x}}{b^{x}}[\/latex].<\/li>\n \t<li>Any number or variable raised to a power of 1 is the number itself.&nbsp;[latex]n^{1}=n[\/latex]<\/li>\n \t<li>Any non-zero number or variable raised to a power of 0 is equal to 1.&nbsp;[latex]n^{0}=1[\/latex]<\/li>\n \t<li>For any nonzero real number [latex]a[\/latex] and natural number [latex]n[\/latex], the negative rule of exponents states that&nbsp;[latex]{a}^{-n}=\\frac{1}{{a}^{n}}[\/latex].<\/li>\n \t<li>With&nbsp;<em>a<\/em>, <em>b<\/em>, <em>m<\/em>, and <em>n<\/em>&nbsp;not equal to zero, and <em>m&nbsp;<\/em>and&nbsp;<em>n<\/em>&nbsp;as integers, the following rules apply:&nbsp;[latex]a^{-m}=\\frac{1}{a^{m}}[\/latex],&nbsp; &nbsp;[latex]\\frac{1}{a^{-m}}=a^{m}[\/latex],&nbsp; &nbsp;[latex]\\frac{a^{-n}}{b^{-m}}=\\frac{b^m}{a^n}[\/latex].<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl>\n \t<dt><strong>absolute value<\/strong><\/dt>\n \t<dd>a number's distance from zero on the number line, which is always positive<\/dd>\n \t<dt><strong>constant<\/strong><\/dt>\n \t<dd>a number whose value always stays the same<\/dd>\n \t<dt><strong>coefficient<\/strong><\/dt>\n \t<dd>a number multiplying a variable<\/dd>\n \t<dt><strong>equation<\/strong><\/dt>\n \t<dd>two expressions connected by an equal sign<\/dd>\n \t<dt><strong>exponent<\/strong><\/dt>\n \t<dd>a number in a superscript position that tells how many times to multiply the base by itself<\/dd>\n \t<dt><strong>expression<\/strong><\/dt>\n \t<dd>a number, a variable, or a combination of numbers and variables and operation symbols<\/dd>\n \t<dt><strong>reciprocal<\/strong><\/dt>\n \t<dd>two fractions are reciprocals if their product is [latex]1[\/latex]<\/dd>\n \t<dt><strong>term<\/strong><\/dt>\n \t<dd>a single number, variable, or a product or quotient of numbers and\/or variables<\/dd>\n \t<dt><strong>variable<\/strong><\/dt>\n \t<dd>a symbol that stands for an unknown quantity, often represented with letters, like <em>x<\/em>, <em>y<\/em>, or <em>z<\/em>.<\/dd>\n<\/dl>\n","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li>The addition property of equality states, for all real numbers [latex]a, b, \\text{ and } c: \\text{ if } a=b \\text{, then } a+c=b+c[\/latex]. That is, we may add or subtract the same amount to both entire sides of an equation without changing its value.<\/li>\n<li>The multiplication property of equality states, for all real numbers [latex]a, b, \\text{ and } c: \\text{ if } a=b \\text{, then } a \\cdot c=b \\cdot c[\/latex]. That is, we may multiply or divide the same amount to both entire sides of an equation without changing its value.<\/li>\n<li>Any point graphed in the coordinate plane is of form [latex]\\left(x, y\\right)[\/latex] where [latex]x[\/latex] is called the x-coordinate and [latex]y[\/latex] is called the y-coordinate. With these coordinates, any point in the plane may be unambiguously located or identified.<\/li>\n<li>The coordinates of any ordered pair contained in the graph of an equation satisfies the equation (makes it a true statement when substituted for x and y).<\/li>\n<li>The slope of a line, of form [latex]m=\\dfrac{\\text{rise}}{\\text{run}}[\/latex], is a measure of the steepness of a line.<\/li>\n<li>Parallel lines have identical slopes; perpendicular lines have opposite, reciprocal slopes.<\/li>\n<\/ul>\n<h2>Key Expressions, Equations, and Inequalities<\/h2>\n<ul>\n<li>[latex]\\displaystyle \\text{Slope }=\\frac{\\text{rise}}{\\text{run}}[\/latex] and [latex]\\displaystyle m=\\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}[\/latex] where [latex]m=\\text{slope}[\/latex]&nbsp;and [latex]\\displaystyle ({{x}_{1}},{{y}_{1}})[\/latex] and [latex]\\displaystyle ({{x}_{2}},{{y}_{2}})[\/latex] are two points on the line.<\/li>\n<li>For any number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>,&nbsp;[latex]\\left(x^{a}\\right)\\left(x^{b}\\right) = x^{a+b}[\/latex].<\/li>\n<li>For any non-zero number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>: [latex]\\displaystyle \\frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[\/latex]<\/li>\n<li>For any positive number <i>x<\/i> and integers <i>a<\/i> and <i>b<\/i>: [latex]\\left(x^{a}\\right)^{b}=x^{a\\cdot{b}}[\/latex].<\/li>\n<li>For any nonzero numbers <i>a<\/i> and <i>b<\/i> and any integer <i>x<\/i>, [latex]\\left(ab\\right)^{x}=a^{x}\\cdot{b^{x}}[\/latex].<\/li>\n<li>For any number <i>a<\/i>, any non-zero number <i>b<\/i>, and any integer <i>x<\/i>, [latex]\\displaystyle {\\left(\\frac{a}{b}\\right)}^{x}=\\frac{a^{x}}{b^{x}}[\/latex].<\/li>\n<li>Any number or variable raised to a power of 1 is the number itself.&nbsp;[latex]n^{1}=n[\/latex]<\/li>\n<li>Any non-zero number or variable raised to a power of 0 is equal to 1.&nbsp;[latex]n^{0}=1[\/latex]<\/li>\n<li>For any nonzero real number [latex]a[\/latex] and natural number [latex]n[\/latex], the negative rule of exponents states that&nbsp;[latex]{a}^{-n}=\\frac{1}{{a}^{n}}[\/latex].<\/li>\n<li>With&nbsp;<em>a<\/em>, <em>b<\/em>, <em>m<\/em>, and <em>n<\/em>&nbsp;not equal to zero, and <em>m&nbsp;<\/em>and&nbsp;<em>n<\/em>&nbsp;as integers, the following rules apply:&nbsp;[latex]a^{-m}=\\frac{1}{a^{m}}[\/latex],&nbsp; &nbsp;[latex]\\frac{1}{a^{-m}}=a^{m}[\/latex],&nbsp; &nbsp;[latex]\\frac{a^{-n}}{b^{-m}}=\\frac{b^m}{a^n}[\/latex].<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl>\n<dt><strong>absolute value<\/strong><\/dt>\n<dd>a number&#8217;s distance from zero on the number line, which is always positive<\/dd>\n<dt><strong>constant<\/strong><\/dt>\n<dd>a number whose value always stays the same<\/dd>\n<dt><strong>coefficient<\/strong><\/dt>\n<dd>a number multiplying a variable<\/dd>\n<dt><strong>equation<\/strong><\/dt>\n<dd>two expressions connected by an equal sign<\/dd>\n<dt><strong>exponent<\/strong><\/dt>\n<dd>a number in a superscript position that tells how many times to multiply the base by itself<\/dd>\n<dt><strong>expression<\/strong><\/dt>\n<dd>a number, a variable, or a combination of numbers and variables and operation symbols<\/dd>\n<dt><strong>reciprocal<\/strong><\/dt>\n<dd>two fractions are reciprocals if their product is [latex]1[\/latex]<\/dd>\n<dt><strong>term<\/strong><\/dt>\n<dd>a single number, variable, or a product or quotient of numbers and\/or variables<\/dd>\n<dt><strong>variable<\/strong><\/dt>\n<dd>a symbol that stands for an unknown quantity, often represented with letters, like <em>x<\/em>, <em>y<\/em>, or <em>z<\/em>.<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-4665\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Authored by<\/strong>: Deborah Devlin. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"Deborah Devlin\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4665","chapter","type-chapter","status-web-only","hentry"],"part":4657,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4665","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4665\/revisions"}],"predecessor-version":[{"id":5403,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4665\/revisions\/5403"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/4657"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4665\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=4665"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=4665"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=4665"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=4665"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}