Lesson Absolute Value equations
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<H2>Absolute Value equations</H2> In this lesson you can learn how to solve simple equations with the unknown under the Absolute Value sign. <H3>Problem 1</H3>Solve absolute value equation {{{abs(x)}}} = {{{2}}}. <B>Solution</B> By the definition, {{{abs(x)}}} = {{{x}}} if {{{x}}} >= {{{0}}}, and {{{abs(x)}}} = {{{-x}}} if {{{x}}} < {{{0}}}. Therefore, we should try to solve two equations. The first equation is {{{x}}} = {{{2}}} at {{{x}}} >= {{{0}}}. The second equation is {{{-x}}} = {{{2}}} at {{{x}}} < {{{0}}}. The first equation has the solution {{{x}}} = {{{2}}}, and it satisfies the condition {{{x}}} >= {{{0}}}. The second equation has the solution {{{x}}} = {{{-2}}}, and it satisfies the condition {{{x}}} < {{{0}}}. So, the given equation has two solutions: {{{x}}} = {{{2}}} and {{{x}}} = {{{-2}}}. You can check the solutions by substituting the found values into the given equation: 1) {{{abs(2)}}} = {{{2}}}, and 2) {{{abs(-2)}}} = {{{2}}}. <H3>Problem 2</H3>Solve absolute value equation {{{abs(x)}}} = {{{5.5}}}. <B>Solution</B> By the definition, {{{abs(x)}}} = {{{x}}} if {{{x}}} >= {{{0}}}, and {{{abs(x)}}} = {{{-x}}} if {{{x}}} < 0. Therefore, we should try to solve two equations. The first equation is {{{x}}} = {{{5.5}}} at {{{x}}} >= {{{0}}}. The second equation is {{{-x}}} = {{{5.5}}} at {{{x}}} < {{{0}}}. The first equation has the solution {{{x}}} = {{{5.5}}}, and it satisfies the condition {{{x}}} >= {{{0}}}. The second equation has the solution {{{x}}} = {{{-5.5}}}, and it satisfies the condition {{{x}}} < {{{0}}}. So, the given equation has two solutions: {{{x}}} = {{{5.5}}} and {{{x}}} = {{{-5.5}}}. You can check the solutions by substituting the found values into the given equation: 1) {{{abs(5.5)}}} = {{{5.5}}}, and 2) {{{abs(-5.5)}}} = {{{5.5}}}. <H3>Problem 3</H3>Solve absolute value equation {{{abs(x)+4}}} = {{{5}}}. <B>Solution</B> Let us simplify the equation first by distracting 4 from both sides: {{{abs(x)}}} = {{{1}}}. Next, by the definition, {{{abs(x)}}} = {{{x}}} if {{{x}}} >= {{{0}}}, and {{{abs(x)}}} = {{{-x}}} if {{{x}}} < {{{0}}}. Therefore, we should try to solve two equations. The first equation is {{{x}}} = {{{1}}} at {{{x}}} >= {{{0}}}. The second equation is {{{-x}}} = {{{1}}} at {{{x}}} < {{{0}}}. The first equation has the solution {{{x}}} = {{{1}}}, and it satisfies the condition {{{x}}} >= {{{0}}}. The second equation has the solution {{{x}}} = {{{-1}}}, and it satisfies the condition {{{x}}} < {{{0}}}. So, the given equation has two solutions: {{{x}}} = {{{1}}} and {{{x}}} = {{{-1}}}. You can check the solutions by substituting the found values into the given equation: 1) {{{abs(1)+4}}} = {{{1+4}}} = {{{5}}}, and 2) {{{abs(-1) +4)}}} = {{{1+4}}} = {{{5}}}. <H3>Problem 4</H3>Solve absolute value equation {{{abs(x+2)}}} = {{{7}}}. <B>Solution</B> By the definition, {{{abs(x+2)}}} = {{{x+2}}} if {{{x+2}}} >= 0, and {{{abs(x+2)}}} = {{{-(x+2)}}} if {{{x+2}}} < 0. Therefore, we should try to solve two equations. The first equation is {{{x+2}}} = {{{7}}} at {{{x}}} >= {{{-2}}}. The second equation is {{{-(x+2)}}} = {{{7}}} at {{{x}}} < {{{-2}}}. The first equation has the solution {{{x}}} = {{{5}}}, and it satisfies the condition {{{x}}} >= {{{-2}}}. The second equation has the solution {{{x}}} = {{{-9}}}, and it satisfies the condition {{{x}}} < {{{-2}}}. So, the given equation has two solutions: {{{x}}} = {{{5}}} and {{{x}}} = {{{-9}}}. You can check the solutions by substituting the found values into the given equation: 1) {{{abs(5+2)}}} = {{{abs(7)}}} = {{{7}}}, and 2) {{{abs(-9+2)}}} = {{{abs(-7)}}} = {{{7}}}. <H3>Problem 5</H3>Solve absolute value equation {{{abs(2x+5)}}} = {{{9}}}. <B>Solution</B> By the definition, {{{abs(2x+5)}}} = {{{2x+5}}} if {{{2x+5}}} >= {{{0}}}, and {{{abs(2x+5)}}} = {{{-(2x+5)}}} if {{{2x+5}}} < {{{0}}}. Therefore, we should try to solve two equations. The first equation is {{{2x+5}}} = {{{9}}} at {{{x}}} >= {{{-5/2}}}. The second equation is {{{-(2x+5)}}} = {{{9}}} at {{{x}}} < {{{-5/2}}}. The first equation has the solution {{{x}}} = {{{2}}}, and it satisfies the condition {{{x}}} >= {{{-5/2}}}. The second equation has the solution {{{x}}} = {{{-7}}}, and it satisfies the condition {{{x}}} < {{{-5/2}}}. So, the given equation has two solutions: {{{x}}} = {{{2}}} and {{{x}}} = {{{-7}}}. You can check the solutions by substituting the found values into the given equation: 1) {{{abs(2*2+5)}}} = {{{abs(9)}}} = {{{9}}}, and 2) {{{abs(2*(-7)+5)}}} = {{{abs(-14+5)}}} = {{{abs(-9)}}} = {{{9}}}. For more complicated equations with the unknown under the Absolute Value sign see the lessons - <A HREF=http://www.algebra.com/algebra/homework/absolute-value/HOW-TO-solve-equations-containing-Linear-Terms-under-Abs-Value-sign-L-1.lesson>HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 1</A>, - <A HREF=http://www.algebra.com/algebra/homework/absolute-value/HOW-TO-solve-equations-containing-Linear-Terms-under-Abs-Value-sign-L-2.lesson>HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 2</A>, - <A HREF=http://www.algebra.com/algebra/homework/absolute-value/HOW-TO-solve-equations-containing-Linear-Terms-under-Abs-Value-sign-L-3.lesson>HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 3</A>, - <A HREF=https://www.algebra.com/algebra/homework/absolute-value/HOW-TO-solve-eqns-containing-Lin-Terms-under-Abs-value-L4.lesson>HOW TO solve equations containing Linear Terms under the Absolute Value sign. Lesson 4</A> - <A HREF=http://www.algebra.com/algebra/homework/absolute-value/HOW-TO-solve-equations-containing-Quadratic-Terms-under-Abs-Value-sign-L-1.lesson>HOW TO solve equations containing Quadratic Terms under the Absolute Value sign. Lesson 1</A> - <A HREF=http://www.algebra.com/algebra/homework/absolute-value/HOW-TO-solve-equations-containing-Quadratic-Terms-under-Abs-Value-sign-L-2.lesson>HOW TO solve equations containing Quadratic Terms under the Absolute Value sign. Lesson 2</A> under the current topic in this site. See also the lesson <A HREF=http://www.algebra.com/algebra/homework/absolute-value/Review-of-lessons-on-Absolute-Value-equations.lesson>OVERVIEW of lessons on Absolute Value equations</A> under the current topic in this site. Use this file/link <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-I.
