Next, we will learn how to solve an absolute value equation. To solve an equation such as [latex]|2x - 6|=8[/latex], we notice that the absolute value will be equal to 8 if the quantity inside the absolute value bars is [latex]8[/latex] or [latex]-8[/latex]. This leads to two different equations we can solve independently. Show [latex]\begin{array}{lll}2x - 6=8\hfill & \text{ or }\hfill & 2x - 6=-8\hfill \\ 2x=14\hfill & \hfill & 2x=-2\hfill \\ x=7\hfill & \hfill & x=-1\hfill \end{array}[/latex] Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point. Equations with one absolute valueA General Note: Absolute Value EquationsThe absolute value of x is written as [latex]|x|[/latex]. It has the following properties: [latex]\begin{array}{l}\text{If } x\ge 0,\text{ then }|x|=x.\hfill \\ \text{If }x<0,\text{ then }|x|=-x.\hfill \end{array}[/latex] For real numbers [latex]A[/latex] and [latex]B[/latex], an equation of the form [latex]|A|=B[/latex], with [latex]B\ge 0[/latex], will have solutions when [latex]A=B[/latex] or [latex]A=-B[/latex]. If [latex]B<0[/latex], the equation [latex]|A|=B[/latex] has no solution. An absolute value equation in the form [latex]|ax+b|=c[/latex] has the following properties: [latex]\begin{array}{l}\text{If }c<0,|ax+b|=c\text{ has no solution}.\hfill \\ \text{If }c=0,|ax+b|=c\text{ has one solution}.\hfill \\ \text{If }c>0,|ax+b|=c\text{ has two solutions}.\hfill \end{array}[/latex] How To: Given an absolute value equation, solve it.
In the next video, we show examples of solving a simple absolute value equation. Example: Solving Absolute Value EquationsSolve the following absolute value equations:
Solution a. [latex]|6x+4|=8[/latex] Write two equations and solve each: [latex]\begin{array}{ll}6x+4\hfill&=8\hfill& 6x+4\hfill&=-8\hfill \\ 6x\hfill&=4\hfill& 6x\hfill&=-12\hfill \\ x\hfill&=\frac{2}{3}\hfill& x\hfill&=-2\hfill \end{array}[/latex] The two solutions are [latex]x=\frac{2}{3}[/latex], [latex]x=-2[/latex]. b. [latex]|3x+4|=-9[/latex] There is no solution as an absolute value cannot be negative. c. [latex]|3x - 5|-4=6[/latex] Isolate the absolute value expression and then write two equations. [latex]\begin{array}{lll}\hfill & |3x - 5|-4=6\hfill & \hfill \\ \hfill & |3x - 5|=10\hfill & \hfill \\ \hfill & \hfill & \hfill \\ 3x - 5=10\hfill & \hfill & 3x - 5=-10\hfill \\ 3x=15\hfill & \hfill & 3x=-5\hfill \\ x=5\hfill & \hfill & x=-\frac{5}{3}\hfill \end{array}[/latex] There are two solutions: [latex]x=5[/latex], [latex]x=-\frac{5}{3}[/latex]. d. [latex]|-5x+10|=0[/latex] The equation is set equal to zero, so we have to write only one equation. [latex]\begin{array}{l}-5x+10\hfill&=0\hfill \\ -5x\hfill&=-10\hfill \\ x\hfill&=2\hfill \end{array}[/latex] There is one solution: [latex]x=2[/latex]. In the two videos that follow, we show examples of how to solve an absolute value equation that requires you to isolate the absolute value first using mathematical operations. Try ItSolve the absolute value equation: [latex]|1 - 4x|+8=13[/latex]. Solution [latex]x=-1[/latex], [latex]x=\frac{3}{2}[/latex] Equations with two absolute valuesSome of our absolute value equations could be of the form [latex]|u|=|v|[/latex] where [latex]u[/latex] and [latex]v[/latex] are algebraic expressions. For example, [latex]|x-3|=|2x+1|[/latex]. How would we solve them? If two algebraic expressions are equal in absolute value, then they are either equal to each other or negatives of each other. The property for absolute value equations says that for any algebraic expression, [latex]u[/latex], and a positive real number, [latex]a[/latex], if [latex]|u|=a[/latex], then [latex]u=a[/latex] or [latex]u=-a[/latex]. This leads us to the following property for equations with two absolute values: Equations with Two Absolute ValuesFor any algebraic expressions, [latex]u[/latex] and [latex]v[/latex], if [latex]|u|=|v|[/latex], then: [latex]u=v[/latex] or [latex]u=-v[/latex]. When we take the opposite of a quantity, we must be careful with the signs and to add parentheses where needed. ExAMPLESolve: [latex]|5x-1|=|2x+3|[/latex]. Show Answer Write the equivalent equations. Solve each equation and check. [latex]\begin{array}{cccc} 5x-1 &=& 2x+3 &or& \,\,\,\, 5x-1 &=& -(2x+3) \\ 5x-1 &=& 2x+3 &or& \,\,\,\, 5x-1 &=& -2x-3 \\ 3x-1 &=& 3 &or& \,\,\,\, 7x-1 &=& -3 \\ 3x &=& 4 &or& \,\,\,\, 7x &=& -2 \\ x &=& \large \frac{4}{3} &or& \,\,\,\, x &=& \large -\frac{2}{7}\end{array}[/latex] AnswerThe two solutions are [latex] x=\frac{4}{3}[/latex] and [latex]x=-\frac{2}{7}[/latex] Try ItAbsolute value equations with no solutionsAs we are solving absolute value equations it is important to be aware of special cases. An absolute value is defined as the distance from 0 on a number line, so it must be a positive number. When an absolute value expression is equal to a negative number, we say the equation has no solution, or DNE. Notice how this happens in the next two examples. ExampleSolve for [latex]x[/latex]. [latex]7+\left|2x-5\right|=4[/latex] Show Solution Notice absolute value is not alone. Subtract [latex]7[/latex] from each side to isolate the absolute value. [latex]\begin{array}{r}7+\left|2x-5\right|=4\,\,\,\,\\\underline{\,-7\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-7\,}\\\left|2x-5\right|=-3\end{array}[/latex] Result of absolute value is negative! The result of an absolute value must always be positive, so we say there is no solution to this equation, or DNE. ExampleSolve for [latex]x[/latex]. [latex]-\frac{1}{2}\left|x+3\right|=6[/latex] Show Solution Notice absolute value is not alone, multiply both sides by the reciprocal of [latex]-\frac{1}{2}[/latex], which is [latex]-2[/latex]. [latex]\begin{array}{r}-\frac{1}{2}\left|x+3\right|=6\,\,\,\,\,\,\,\,\,\,\,\,\\\,\,\,\,\,\,\,\,\left(-2\right)-\frac{1}{2}\left|x+3\right|=\left(-2\right)6\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left|x+3\right|=-12\,\,\,\,\,\end{array}[/latex] Again, we have a result where an absolute value is negative! There is no solution to this equation, or DNE. In this last video, we show more examples of absolute value equations that have no solutions. (6.3.2) – Solve inequalities containing absolute valuesLet’s apply what you know about solving equations that contain absolute values and what you know about inequalities to solve inequalities that contain absolute values. Let’s start with a simple inequality. [latex]\left|x\right|\leq 4[/latex] This inequality is read, “the absolute value of [latex]x[/latex] is less than or equal to 4.” If you are asked to solve for [latex]x[/latex], you want to find out what values of [latex]x[/latex] are 4 units or less away from 0 on a number line. You could start by thinking about the number line and what values of [latex]x[/latex] would satisfy this equation. 4 and [latex]−4[/latex] are both four units away from 0, so they are solutions. 3 and [latex]−3[/latex] are also solutions because each of these values is less than 4 units away from 0. So are 1 and [latex]−1[/latex], 0.5 and [latex]−0.5[/latex], and so on—there are an infinite number of values for [latex]x[/latex] that will satisfy this inequality. The graph of this inequality will have two closed circles, at 4 and [latex]−4[/latex]. The distance between these two values on the number line is colored in blue because all of these values satisfy the inequality. The solution can be written this way: Inequality: [latex]-4\leq x\leq4[/latex] Interval: [latex]\left[-4,4\right][/latex] The situation is a little different when the inequality sign is “greater than” or “greater than or equal to.” Consider the simple inequality [latex]\left|x\right|>3[/latex]. Again, you could think of the number line and what values of [latex]x[/latex] are greater than 3 units away from zero. This time, 3 and [latex]−3[/latex] are not included in the solution, so there are open circles on both of these values. 2 and [latex]−2[/latex] would not be solutions because they are not more than 3 units away from 0. But 5 and [latex]−5[/latex] would work, and so would all of the values extending to the left of [latex]−3[/latex] and to the right of 3. The graph would look like the one below. The solution to this inequality can be written this way: Inequality: [latex]x<−3[/latex] or [latex]x>3[/latex]. Interval: [latex]\left(-\infty, -3\right)\cup\left(3,\infty\right)[/latex] In the following video, you will see examples of how to solve and express the solution to absolute value inequalities involving both AND and OR. Writing Solutions to Absolute Value InequalitiesFor any positive value of [latex]a[/latex] and [latex]x[/latex], a single variable, or any algebraic expression: Absolute Value InequalityEquivalent InequalityInterval Notation[latex]\left|{ x }\right|\le{ a}[/latex][latex]{ -a}\le{x}\le{ a}[/latex][latex]\left[-a, a\right][/latex][latex]\left| x \right|\lt{a}[/latex][latex]{ -a}\lt{x}\lt{ a}[/latex][latex]\left(-a, a\right)[/latex][latex]\left| x \right|\ge{ a}[/latex][latex]{x}\le\text{−a}[/latex] or [latex]{x}\ge{ a}[/latex] [latex]\left(-\infty,-a\right]\cup\left[a,\infty\right)[/latex][latex]\left| x \right|\gt\text{a}[/latex][latex]\displaystyle{x}\lt\text{−a}[/latex] or [latex]{x}\gt{ a}[/latex] [latex]\left(-\infty,-a\right)\cup\left(a,\infty\right)[/latex]Let’s look at a few more examples of inequalities containing absolute values. ExampleSolve for [latex]x[/latex]. [latex]\left|x+3\right|\gt4[/latex] Show Solution Since this is a “greater than” inequality, the solution can be rewritten according to the “greater than” rule. [latex] \displaystyle x+3<-4\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\,\,x+3>4[/latex] Solve each inequality. [latex]\begin{array}{r}x+3<-4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x+3>4\\\underline{\,\,\,\,-3\,\,\,\,\,-3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{\,\,\,\,\,\,-3\,\,-3}\\x\,\,\,\,\,\,\,\,\,<-7\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\,\,\,\,\,\,\,\,\,>1\\\\x<-7\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\,x>1\,\,\,\,\,\,\,\end{array}[/latex] Check the solutions in the original equation to be sure they work. Check the end point of the first related equation, [latex]−7[/latex] and the end point of the second related equation, 1. [latex] \displaystyle \begin{array}{r}\,\,\,\left| x+3 \right|>4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| x+3 \right|>4\\\left| -7+3 \right|=4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| 1+3 \right|=4\\\,\,\,\,\,\,\,\left| -4 \right|=4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| 4 \right|=4\\\,\,\,\,\,\,\,\,\,\,\,\,4=4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4=4\end{array}[/latex] Try [latex]−10[/latex], a value less than [latex]−7[/latex], and 5, a value greater than 1, to check the inequality. [latex] \displaystyle \begin{array}{r}\,\,\,\,\,\left| x+3 \right|>4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| x+3 \right|>4\\\left| -10+3 \right|>4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| 5+3 \right|>4\\\,\,\,\,\,\,\,\,\,\,\left| -7 \right|>4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| 8 \right|>4\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,7>4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,8>4\end{array}[/latex] Both solutions check! AnswerInequality: [latex] \displaystyle x<-7\,\,\,\,\,\text{or}\,\,\,\,\,x>1[/latex] Interval: [latex]\left(-\infty, -7\right)\cup\left(1,\infty\right)[/latex] Graph: ExampleSolve for [latex]y[/latex]. [latex] \displaystyle 3\left|2y+6\right|-9<27[/latex] Show Solution Begin to isolate the absolute value by adding 9 to both sides of the inequality. [latex] \displaystyle \begin{array}{r}3\left| 2y+6 \right|-9<27\\\underline{\,\,+9\,\,\,+9}\\3\left| 2y+6 \right|\,\,\,\,\,\,\,\,<36\end{array}[/latex] Divide both sides by 3 to isolate the absolute value. [latex]\begin{array}{r}\underline{3\left| 2y+6 \right|}\,<\underline{36}\\3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,3\,\\\,\,\,\,\,\,\,\,\,\left| 2y+6 \right|<12\end{array}[/latex] Write the absolute value inequality using the “less than” rule. Subtract 6 from each part of the inequality. [latex]\begin{array}{r}-12<2y+6<12\\\underline{\,\,-6\,\,\,\,\,\,\,\,\,\,\,\,\,-6\,\,\,-6}\\-18\,<\,2y\,\,\,\,\,\,\,\,\,<\,\,6\,\end{array}[/latex] Divide by 2 to isolate the variable. [latex]\begin{array}{r}\underline{-18}<\underline{2y}<\underline{\,6\,}\\2\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,2\,\,\\-9<\,\,y\,\,\,\,<\,3\end{array}[/latex] AnswerInequality: [latex] \displaystyle -9<\,\,y\,\,<3[/latex] Interval: [latex]\left(-9,3\right)[/latex] Graph: In the following video, you will see an example of solving multi-step absolute value inequalities involving an OR situation. Identify cases of inequalities containing absolute values that have no solutionsAs with equations, there may be instances in which there is no solution to an inequality. ExampleSolve for [latex]x[/latex]. [latex]\left|2x+3\right|+9\leq 7[/latex] Show Solution Isolate the absolute value by subtracting 9 from both sides of the inequality. [latex] \displaystyle \begin{array}{r}\left| 2x+3 \right|+9\,\le \,\,\,7\,\,\\\underline{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-9\,\,\,\,\,-9}\\\,\,\,\,\,\,\,\left| 2x+3 \right|\,\,\,\le -2\,\end{array}[/latex] The absolute value of a quantity can never be a negative number, so there is no solution to the inequality. AnswerNo solution SummaryAbsolute inequalities can be solved by rewriting them using compound inequalities. The first step to solving absolute inequalities is to isolate the absolute value. The next step is to decide whether you are working with an OR inequality or an AND inequality. If the inequality is greater than a number, we will use OR. If the inequality is less than a number, we will use AND. Remember that if we end up with an absolute value greater than or less than a negative number, there is no solution. |