The absolute number of a number a is written as Show
$$\left | a \right |$$ And represents the distance between a and 0 on a number line. An absolute value equation is an equation that contains an absolute value expression. The equation $$\left | x \right |=a$$ Has two solutions x = a and x = -a because both numbers are at the distance a from 0. To solve an absolute value equation as $$\left | x+7 \right |=14$$ You begin by making it into two separate equations and then solving them separately. $$x+7 =14$$ $$x+7\, {\color{green} {-\, 7}}\, =14\, {\color{green} {-\, 7}}$$ $$x=7$$ or $$x+7 =-14$$ $$x+7\, {\color{green} {-\, 7}}\, =-14\, {\color{green} {-\, 7}}$$ $$x=-21$$ An absolute value equation has no solution if the absolute value expression equals a negative number since an absolute value can never be negative. The inequality $$\left | x \right |<2$$ Represents the distance between x and 0 that is less than 2 Whereas the inequality $$\left | x \right |>2$$ Represents the distance between x and 0 that is greater than 2 You can write an absolute value inequality as a compound inequality. $$\left | x \right |<2\: or $$-2<x<2$$ This holds true for all absolute value inequalities. $$\left | ax+b \right |<c,\: where\: c>0$$ $$=-c<ax+b<c$$ $$\left | ax+b \right |>c,\: where\: c>0$$ $$=ax+b<-c\: or\: ax+b>c$$ You can replace > above with ≥ and < with ≤. When solving an absolute value inequality it's necessary to first isolate the absolute value expression on one side of the inequality before solving the inequality. Example Solve the absolute value inequality $$2\left |3x+9 \right |<36$$ $$\frac{2\left |3x+9 \right |}{2}<\frac{36}{2}$$ $$\left | 3x+9 \right |<18$$ $$-18<3x+9<18$$ $$-18\, {\color{green} {-\, 9}}<3x+9\, {\color{green} {-\, 9}}<18\, {\color{green} {-\, 9}}$$ $$-27<3x<9$$ $$\frac{-27}{{\color{green} 3}}<\frac{3x}{{\color{green} 3}}<\frac{9}{{\color{green} 3}}$$ $$-9<x<3$$ Video lessonSolve the absolute value equation $$4 \left |2x -1 \right | -2 = 10$$ Learning Objectives
(6.3.1) – Solve equations containing absolute valuesNext, 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. [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:
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]. 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]. 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] ExampleSolve for [latex]x[/latex]. [latex]-\frac{1}{2}\left|x+3\right|=6[/latex] 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:
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] ExampleSolve for [latex]y[/latex]. [latex] \displaystyle 3\left|2y+6\right|-9<27[/latex] 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] 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. |