{"id":1729,"date":"2021-08-27T19:39:47","date_gmt":"2021-08-27T19:39:47","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1729"},"modified":"2022-02-01T20:05:48","modified_gmt":"2022-02-01T20:05:48","slug":"summary-review-6","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/summary-review-6\/","title":{"raw":"Summary: Review","rendered":"Summary: Review"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li aria-level=\"1\">Properties of real numbers:\u00a0For any real numbers <em>a, b,<\/em> and <em>c<\/em>,<\/li>\r\n<\/ul>\r\n<table summary=\"A table with six rows and three columns. The first entry of the first row is blank while the remaining columns read: Addition and Multiplication. The first entry of the second row reads: Commutative Property. The second column entry reads a plus b equals b plus a. The third column entry reads a times b equals b times a. The first entry of the third row reads Associative Property. The second column entry reads: a plus the quantity b plus c in parenthesis equals the quantity a plus b in parenthesis plus c. The third column entry reads: a times the quantity b times c in parenthesis equals the quantity a times b in parenthesis times c. The first entry of the fourth row reads: Distributive Property. The second and third column are combined on this row and read: a times the quantity b plus c in parenthesis equals a times b plus a times c. The first entry in the fifth row reads: Identity Property. The second column entry reads: There exists a unique real number called the additive identity, 0, such that for any real number a, a + 0 = a. The third column entry reads: There exists a unique real number called the multiplicative inverse, 1, such that for any real number a, a times 1 equals a. The first entry in the sixth row reads: Inverse Property. The second column entry reads: Every real number a has an additive inverse, or opposite, denoted negative a such that, a plus negative a equals zero. The third column entry reads: Every nonzero real\">\r\n<tbody>\r\n<tr>\r\n<th><\/th>\r\n<th>Addition<\/th>\r\n<th>Multiplication<\/th>\r\n<\/tr>\r\n<\/tbody>\r\n<tbody>\r\n<tr>\r\n<td><strong>Commutative Property<\/strong><\/td>\r\n<td>[latex]a+b=b+a[\/latex]<\/td>\r\n<td>[latex]a\\cdot b=b\\cdot a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Associative Property<\/strong><\/td>\r\n<td>[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/td>\r\n<td>[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Distributive Property<\/strong><\/td>\r\n<td>[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Identity Property<\/strong><\/td>\r\n<td>There exists a unique real number called the additive identity, 0, such that, for any real number <em>a<\/em>\r\n<div style=\"text-align: center;\">[latex]a+0=a[\/latex]<\/div><\/td>\r\n<td>There exists a unique real number called the multiplicative identity, 1, such that, for any real number <em>a<\/em>\r\n<div style=\"text-align: center;\">[latex]a\\cdot 1=a[\/latex]<\/div><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Inverse Property<\/strong><\/td>\r\n<td>Every real number a has an additive inverse, or opposite, denoted [latex]\u2013a[\/latex], such that\r\n<div style=\"text-align: center;\">[latex]a+\\left(-a\\right)=0[\/latex]<\/div><\/td>\r\n<td>Every nonzero real number <em>a<\/em> has a multiplicative inverse, or reciprocal, denoted [latex]\\Large\\frac{1}{a}[\/latex], such that\r\n<div style=\"text-align: center;\">[latex]a\\cdot \\left(\\Large\\frac{1}{a}\\normalsize\\right)=1[\/latex]<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">We simplify an expression by removing grouping symbols and combining like terms.<\/li>\r\n \t<li>Properties of Equality For two expressions <em>S<\/em> and <em>T<\/em> and any constant <em>c<\/em>,\r\n<ul>\r\n \t<li><strong>Addition Property of Equality:<\/strong> If [latex] S=T[\/latex] then [latex]S+c=T+c[\/latex]<\/li>\r\n \t<li><strong>Multiplication Property of Equality:<\/strong> If [latex]S=T[\/latex] then [latex]S \\cdot c = T \\cdot c[\/latex], provided [latex]c \\neq 0[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">To solve a multi-step equation\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">Multiply to clear any fractions or decimals (optional)<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">Simplify each side by clearing parentheses and combining like terms.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">Add or subtract to isolate the variable term\u2014possibly a term with the variable.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"2\">Multiply or divide to isolate the variable.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The the solutions of [latex]|x|=a[\/latex] is [latex]x = a[\/latex] or [latex]x = -a[\/latex]<\/li>\r\n \t<li>For any positive value of <i>a\u00a0<\/i>and\u00a0<em>x,<\/em>\u00a0a single variable, or any algebraic expression:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Absolute Value Inequality<\/strong><\/td>\r\n<td><strong>Equivalent Inequality<\/strong><\/td>\r\n<td><strong>Interval Notation<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left|{ x }\\right|\\le{ a}[\/latex]<\/td>\r\n<td>[latex]{ -a}\\le{x}\\le{ a}[\/latex]<\/td>\r\n<td>[latex]\\left[-a, a\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left| x \\right|\\lt{a}[\/latex]<\/td>\r\n<td>[latex]{ -a}\\lt{x}\\lt{ a}[\/latex]<\/td>\r\n<td>[latex]\\left(-a, a\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left| x \\right|\\ge{ a}[\/latex]<\/td>\r\n<td>[latex]{x}\\le\\text{\u2212a}[\/latex] or [latex]{x}\\ge{ a}[\/latex]<\/td>\r\n<td>\u00a0[latex]\\left(-\\infty,-a\\right]\\cup\\left[a,\\infty\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left| x \\right|\\gt\\text{a}[\/latex]<\/td>\r\n<td>[latex]\\displaystyle{x}\\lt\\text{\u2212a}[\/latex]\u00a0or [latex]{x}\\gt{ a}[\/latex]<\/td>\r\n<td>\u00a0[latex]\\left(-\\infty,-a\\right)\\cup\\left(a,\\infty\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\"><strong>coefficient:<\/strong> constant factor in a term<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\"><strong>like terms:<\/strong> have exactly the same variable factors<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\"><strong>absolute value:<\/strong>\u00a0the absolute value of a number [latex]n[\/latex], written [latex]|n|[\/latex], is its distance from [latex]0[\/latex] on the number line. [latex]|n| \\geq 0[\/latex] for every real number [latex]n[\/latex].<\/li>\r\n<\/ul>","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li aria-level=\"1\">Properties of real numbers:\u00a0For any real numbers <em>a, b,<\/em> and <em>c<\/em>,<\/li>\n<\/ul>\n<table summary=\"A table with six rows and three columns. The first entry of the first row is blank while the remaining columns read: Addition and Multiplication. The first entry of the second row reads: Commutative Property. The second column entry reads a plus b equals b plus a. The third column entry reads a times b equals b times a. The first entry of the third row reads Associative Property. The second column entry reads: a plus the quantity b plus c in parenthesis equals the quantity a plus b in parenthesis plus c. The third column entry reads: a times the quantity b times c in parenthesis equals the quantity a times b in parenthesis times c. The first entry of the fourth row reads: Distributive Property. The second and third column are combined on this row and read: a times the quantity b plus c in parenthesis equals a times b plus a times c. The first entry in the fifth row reads: Identity Property. The second column entry reads: There exists a unique real number called the additive identity, 0, such that for any real number a, a + 0 = a. The third column entry reads: There exists a unique real number called the multiplicative inverse, 1, such that for any real number a, a times 1 equals a. The first entry in the sixth row reads: Inverse Property. The second column entry reads: Every real number a has an additive inverse, or opposite, denoted negative a such that, a plus negative a equals zero. The third column entry reads: Every nonzero real\">\n<tbody>\n<tr>\n<th><\/th>\n<th>Addition<\/th>\n<th>Multiplication<\/th>\n<\/tr>\n<\/tbody>\n<tbody>\n<tr>\n<td><strong>Commutative Property<\/strong><\/td>\n<td>[latex]a+b=b+a[\/latex]<\/td>\n<td>[latex]a\\cdot b=b\\cdot a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Associative Property<\/strong><\/td>\n<td>[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/td>\n<td>[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Distributive Property<\/strong><\/td>\n<td>[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><strong>Identity Property<\/strong><\/td>\n<td>There exists a unique real number called the additive identity, 0, such that, for any real number <em>a<\/em><\/p>\n<div style=\"text-align: center;\">[latex]a+0=a[\/latex]<\/div>\n<\/td>\n<td>There exists a unique real number called the multiplicative identity, 1, such that, for any real number <em>a<\/em><\/p>\n<div style=\"text-align: center;\">[latex]a\\cdot 1=a[\/latex]<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td><strong>Inverse Property<\/strong><\/td>\n<td>Every real number a has an additive inverse, or opposite, denoted [latex]\u2013a[\/latex], such that<\/p>\n<div style=\"text-align: center;\">[latex]a+\\left(-a\\right)=0[\/latex]<\/div>\n<\/td>\n<td>Every nonzero real number <em>a<\/em> has a multiplicative inverse, or reciprocal, denoted [latex]\\Large\\frac{1}{a}[\/latex], such that<\/p>\n<div style=\"text-align: center;\">[latex]a\\cdot \\left(\\Large\\frac{1}{a}\\normalsize\\right)=1[\/latex]<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">We simplify an expression by removing grouping symbols and combining like terms.<\/li>\n<li>Properties of Equality For two expressions <em>S<\/em> and <em>T<\/em> and any constant <em>c<\/em>,\n<ul>\n<li><strong>Addition Property of Equality:<\/strong> If [latex]S=T[\/latex] then [latex]S+c=T+c[\/latex]<\/li>\n<li><strong>Multiplication Property of Equality:<\/strong> If [latex]S=T[\/latex] then [latex]S \\cdot c = T \\cdot c[\/latex], provided [latex]c \\neq 0[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">To solve a multi-step equation\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"2\">Multiply to clear any fractions or decimals (optional)<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">Simplify each side by clearing parentheses and combining like terms.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">Add or subtract to isolate the variable term\u2014possibly a term with the variable.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"2\">Multiply or divide to isolate the variable.<\/li>\n<\/ul>\n<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The the solutions of [latex]|x|=a[\/latex] is [latex]x = a[\/latex] or [latex]x = -a[\/latex]<\/li>\n<li>For any positive value of <i>a\u00a0<\/i>and\u00a0<em>x,<\/em>\u00a0a single variable, or any algebraic expression:<br \/>\n<table>\n<tbody>\n<tr>\n<td><strong>Absolute Value Inequality<\/strong><\/td>\n<td><strong>Equivalent Inequality<\/strong><\/td>\n<td><strong>Interval Notation<\/strong><\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left|{ x }\\right|\\le{ a}[\/latex]<\/td>\n<td>[latex]{ -a}\\le{x}\\le{ a}[\/latex]<\/td>\n<td>[latex]\\left[-a, a\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left| x \\right|\\lt{a}[\/latex]<\/td>\n<td>[latex]{ -a}\\lt{x}\\lt{ a}[\/latex]<\/td>\n<td>[latex]\\left(-a, a\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left| x \\right|\\ge{ a}[\/latex]<\/td>\n<td>[latex]{x}\\le\\text{\u2212a}[\/latex] or [latex]{x}\\ge{ a}[\/latex]<\/td>\n<td>\u00a0[latex]\\left(-\\infty,-a\\right]\\cup\\left[a,\\infty\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left| x \\right|\\gt\\text{a}[\/latex]<\/td>\n<td>[latex]\\displaystyle{x}\\lt\\text{\u2212a}[\/latex]\u00a0or [latex]{x}\\gt{ a}[\/latex]<\/td>\n<td>\u00a0[latex]\\left(-\\infty,-a\\right)\\cup\\left(a,\\infty\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><strong>coefficient:<\/strong> constant factor in a term<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><strong>like terms:<\/strong> have exactly the same variable factors<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><strong>absolute value:<\/strong>\u00a0the absolute value of a number [latex]n[\/latex], written [latex]|n|[\/latex], is its distance from [latex]0[\/latex] on the number line. [latex]|n| \\geq 0[\/latex] for every real number [latex]n[\/latex].<\/li>\n<\/ul>\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-1729\">\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>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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 9: Real Numbers, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/a>. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra: Using Properties of Real Numbers. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakerintermediatealgebra\/chapter\/read-use-properties-of-real-numbers\/\">https:\/\/courses.lumenlearning.com\/waymakerintermediatealgebra\/chapter\/read-use-properties-of-real-numbers\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction\">https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction<\/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":169134,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"College Algebra: Using Properties of Real Numbers\",\"author\":\"\",\"organization\":\"\",\"url\":\"https:\/\/courses.lumenlearning.com\/waymakerintermediatealgebra\/chapter\/read-use-properties-of-real-numbers\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 9: Real Numbers, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1729","chapter","type-chapter","status-publish","hentry"],"part":256,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1729","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":5,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1729\/revisions"}],"predecessor-version":[{"id":3658,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1729\/revisions\/3658"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/256"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1729\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1729"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1729"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1729"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1729"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}