Absolute Value Equations: Videos & Practice Problems

Solving absolute value equations involves isolating the absolute value expression and rewriting $\mathrm{|x|}=a$ as two separate equations: $x=a$ and $x=-a$. When $a$ is zero, the solution is where the expression inside equals zero; if $a$ is negative, no solution exists. For equations with two absolute values, set the inner expressions equal and also equal to negatives of each other. This approach enhances understanding of polynomial terms, coefficients, and expressions in standard form.

Solving Absolute Value Equations

The absolute value of a number represents its distance from zero on the number line, regardless of direction. For example, the absolute value of 2 is 2, and the absolute value of -2 is also 2, since both are two units away from zero. This fundamental concept helps in solving absolute value equations, which may initially seem complex but can be simplified using this distance interpretation.

When solving an equation involving absolute value, such as \(|x| = 2\), the goal is to find all values of \(x\) whose distance from zero is 2. This means \(x\) can be either 2 or -2, since both satisfy the equation. Substituting these values back into the original equation confirms they are valid solutions. Thus, the solution set is \(\{2, -2\}\).

This example illustrates a general rule for absolute value equations: if \(|x| = a\) where \(a\) is a positive number, then \(x = a\) or \(x = -a\). This rule applies not only when the absolute value contains a single variable but also when it contains an expression. For instance, in the equation \(|x + 1| = 2\), the expression inside the absolute value, \(x + 1\), can equal 2 or -2. Solving these two linear equations separately, \(x + 1 = 2\) and \(x + 1 = -2\), yields \(x = 1\) and \(x = -3\) respectively. Both values satisfy the original equation, so the solution set is \(\{1, -3\}\).

It is important to isolate the absolute value expression on one side of the equation before applying this rule. For example, to solve \(|x + 1| + 3 = 5\), first subtract 3 from both sides to isolate the absolute value: \(|x + 1| = 2\). Then apply the rule to find the solutions as before.

In summary, solving absolute value equations involves isolating the absolute value expression and then rewriting the equation \(|expression| = a\) as two separate equations: \(expression = a\) and \(expression = -a\). Solving these linear equations provides all possible solutions. This method is effective when \(a\) is a positive number, and further considerations are needed when \(a\) is zero or negative.

Solving Absolute Value Equations Special Cases

When solving equations involving the absolute value expression, such as |x| = a, the value of a plays a crucial role in determining the number and type of solutions. The absolute value represents the distance from zero, which is always non-negative. Therefore, when a is positive, the equation |x| = a can be rewritten as two separate linear equations: x = a and x = -a. Solving these gives two solutions because the distance from zero can be achieved in two directions on the number line.

However, when a = 0, the equation |x| = 0 means the distance from zero is zero, which only occurs if the expression inside the absolute value is exactly zero. This simplifies the problem to a single equation without absolute value: x = 0. Thus, there is exactly one solution in this case.

In contrast, if a is negative, such as in |x| = -2, the equation has no solution. Since distance cannot be negative, the absolute value expression cannot equal a negative number. This results in a contradiction, meaning the equation is always false regardless of the value of x. Therefore, no solutions exist when a is less than zero.

Understanding these distinctions is essential for solving absolute value equations effectively. The key takeaway is that the absolute value equation |x| = a has two solutions if a > 0, one solution if a = 0, and no solutions if a < 0. This knowledge allows for accurate interpretation and solution of absolute value problems in algebra.

Solve the following equations.
(A) $2\mid x\mid =10$

• A. $x=\{20,-20\}$
• B. $x=\{12,-12\}$
• C. $x=\{10,-10\}$
• D. $x=\{5,-5\}$

Answer: (D)

Solve the following equations.
(B) $\mid 3x-5\mid -6=2$

• A. $x=\{\frac{13}{3},-1\}$
• B. $x=\{-\frac{13}{3},1\}$
• C. $x=\{3,-\frac{1}{3}\}$
• D. No solution

Answer: (A)

Solve the following equations.
(C) $0.4\mid 3y+2\mid =3.6$

• A. $y=\{14,-\frac{38}{3}\}$
• B. $y=\{\frac{7}{3},-\frac{11}{3}\}$
• C. $y=\{-\frac{7}{3},\frac{11}{3}\}$
• D. $y=\{-14,\frac{38}{3}\}$

Answer: (B)

Solve the following equations with absolute value.
(A) $3\mid 4x+1\mid +7=2$

• A. $x=\{\frac{1}{2},-1\}$
• B. $x=\{-\frac{1}{6},\frac{2}{3}\}$
• C. $x=\{\frac{1}{6},-\frac{2}{3}\}$
• D. No solution

Answer: (D)

Solve the following equations with absolute value.
(B) $5\mid 6x-9\mid +3=3$

• A. $x=-\frac{3}{2}$
• B. $x=0$
• C. $x=\frac{3}{2}$
• D. $x=3$

Answer: (C)

Solving Equations Involving 2 Absolute Values

Solving equations involving absolute values requires understanding how to handle the expressions inside the absolute value symbols. When dealing with a single absolute value equation, the goal is to isolate the absolute value expression on one side, resulting in a form like |x + 1| = 2. This allows the equation to be rewritten as two separate linear equations by setting the expression inside equal to both the positive and negative values of the number outside the absolute value. For example, x + 1 = 2 and x + 1 = -2 lead to solutions x = 1 and x = -3, respectively. These solutions are then combined into a solution set, denoted by curly brackets.

When an equation contains two absolute values set equal to each other, such as |x + 1| = |2x - 4|, a similar approach applies. The key is to recognize that if |A| = |B|, then either A = B or A = -B. This principle allows the original equation to be split into two separate equations without absolute values: one where the expressions inside the absolute values are equal, and another where one expression equals the negative of the other.

Applying this to the example, the first equation is x + 1 = 2x - 4, and the second is x + 1 = -(2x - 4). Solving the first equation involves isolating the variable by adding 4 to both sides, yielding x + 5 = 2x, then subtracting x from both sides to get 5 = x. The second equation requires distributing the negative sign: x + 1 = -2x + 4. Subtracting 1 from both sides gives x = -2x + 3, and adding 2x to both sides results in 3x = 3. Dividing both sides by 3 yields x = 1. Thus, the solution set for the original equation is {1, 5}.

This method highlights the importance of understanding absolute value properties and linear equation solving techniques. By transforming absolute value equations into equivalent linear forms, one can systematically find all possible solutions. This approach is essential for mastering absolute value equations, especially when multiple absolute values are involved, and reinforces foundational algebraic skills.

Solve the following equations with 2 absolute values.
(A) $\mid 3x+4\mid =\mid -2x+7\mid$

• A. $x=\{\frac{3}{5},-11\}$
• B. $x=\{3,-11\}$
• C. $x=\{\frac{3}{5},3\}$
• D. $x=\{-\frac{3}{5},3\}$

Answer: (A)

Solve the following equations with 2 absolute values.
(B) $\mid \frac{2x-3}{4}\mid =\mid x+\frac{1}{2}\mid$

• A. $x=\{-\frac{5}{2},\frac{5}{6}\}$
• B. $x=\{\frac{5}{2},-\frac{1}{6}\}$
• C. $x=\{-\frac{5}{2},\frac{1}{6}\}$
• D. $x=\{\frac{5}{2},-\frac{5}{6}\}$

Answer: (C)

Solving Equations Involving 2 Absolute Values Example 1

To solve the equation involving absolute values, such as $$|3x\; +\; 4|\; =\; |-8|$$, start by simplifying the absolute value on the right side. Since the absolute value of a number represents its distance from zero, $$|-8|\; =\; 8$$. This reduces the equation to $$|3x\; +\; 4|\; =\; 8$$. When an absolute value equals a positive number, the expression inside the absolute value can be either the positive or negative of that number. Therefore, split the equation into two cases: $$3x\; +\; 4\; =\; 8$$ and $$3x\; +\; 4\; =\; -8$$.

For the first case, isolate the variable term by subtracting 4 from both sides: $$3x\; =\; 8\; -\; 4\; =\; 4$$. Then, divide both sides by 3 to solve for $x$: $$x\; =\; \backslash frac\{4\}\{3\}$$.

For the second case, similarly subtract 4 from both sides: $$3x\; =\; -8\; -\; 4\; =\; -12$$. Dividing both sides by 3 gives $$x\; =\; \backslash frac\{-12\}\{3\}\; =\; -4$$.

Thus, the solutions to the equation are $$x\; =\; \backslash frac\{4\}\{3\}$$ and $$x\; =\; -4$$. These values satisfy the original absolute value equation, demonstrating how to handle equations where an absolute value expression equals a positive constant by considering both positive and negative scenarios of the inner expression.

Solving Equations Involving 2 Absolute Values Example 2

To solve the equation involving absolute values, |2x − 4| = |−2x + 3|, we use the property that if two absolute values are equal, then either the expressions inside are equal or they are negatives of each other. This leads to two separate equations to solve.

First, set the expressions inside the absolute values equal to each other:

\$2x - 4 = -2x + 3\(

Rearranging terms to isolate the variable, add 4 to both sides:

\)2x = -2x + 7\(

Then add \)2x\( to both sides:

\)4x = 7\(

Dividing both sides by 4 gives the solution:

\)x = \frac{7}{4}\(

Next, consider the case where one expression equals the negative of the other:

\)2x - 4 = -(-2x + 3)\(

Distribute the negative sign on the right side:

\)2x - 4 = 2x - 3\(

Add 4 to both sides:

\)2x = 2x + 1\(

Subtract \)2x\( from both sides:

\)0 = 1\(

This is a contradiction, indicating no solution exists for this case.

Therefore, the only solution to the original equation is:

\)x = \frac{7}{4}$

This approach highlights the importance of understanding absolute value equations and how to break them into linear cases. By carefully isolating variables and checking for contradictions, we ensure all valid solutions are found.

Solving Equations Involving 2 Absolute Values Example 3

To solve the equation involving two absolute values on the same side, such as |x - 3| - |3x - 9| = 0, the first step is to isolate one absolute value expression on one side. By adding |3x - 9| to both sides, the equation becomes |x - 3| = |3x - 9|, which is a standard form where two absolute values are set equal.

When two absolute values are equal, the expressions inside can either be equal or negatives of each other. This leads to two separate equations to solve:

1. x - 3 = 3x - 9

2. x - 3 = -(3x - 9)

For the first equation, rearranging terms involves adding 9 to both sides and subtracting x from both sides:

\(x - 3 = 3x - 9\)

Add 9 to both sides:

\(x - 3 + 9 = 3x - 9 + 9 \implies x + 6 = 3x\)

Subtract x from both sides:

\$6 = 2x\(

Divide both sides by 2:

\)x = 3\(

For the second equation, distribute the negative sign and then isolate x:

\)x - 3 = -3x + 9\(

Add 3 to both sides:

\)x = -3x + 12\(

Add 3x to both sides:

\)4x = 12\(

Divide both sides by 4:

\)x = 3$

Both equations yield the same solution, x = 3. This means the original equation has a single solution, which is 3. Understanding how to manipulate absolute value equations by isolating terms and considering both positive and negative cases is essential for solving these types of problems effectively.

Here’s what students ask on this topic:

How do you solve an absolute value equation when the absolute value is isolated?

To solve an absolute value equation where the absolute value is isolated, such as $|x|=a$ with $a$ positive, you rewrite it as two separate equations: $x=a$ and $x=-a$. This is because absolute value represents distance from zero, so the expression inside the absolute value can be either positive or negative $a$. After rewriting, solve each equation separately to find the solution set. For example, if $|x|=3$, then $x=3$ or $x=-3$. Always ensure the absolute value is isolated before applying this rule.

What is the solution to an absolute value equation when the value equals zero?

When solving an absolute value equation like $|x|=0$, the only solution is where the expression inside the absolute value equals zero. This is because the absolute value represents distance, and the only way for the distance to be zero is if the expression itself is zero. So, you set the inside equal to zero: $x=0$. This gives a single solution. For example, if you have $|x+1|=0$, then solve $x+1=0$ to get $x=-1$.

Why does an absolute value equation have no solution when the absolute value equals a negative number?

An absolute value equation has no solution when the absolute value equals a negative number because absolute value represents distance, and distance cannot be negative. For example, if you have $|x|=-2$, this means the distance between $x$ and zero is negative two, which is impossible. Therefore, no value of $x$ satisfies this equation, and the solution set is empty. This rule applies to any equation of the form $|x|=a$ where $a$ is negative.

How do you solve an equation with two absolute values set equal to each other?

When solving an equation with two absolute values equal, such as $|f\left(x\right)|=|g\left(x\right)|$, you rewrite it as two separate equations without absolute values: $f\left(x\right)=g\left(x\right)$ and $f\left(x\right)=-g\left(x\right)$. Then solve each equation separately. For example, if $|x+1|=|2x-4|$, set $x+1=2x-4$ and $x+1=-(2x-4)$, then solve both for $x$. This method helps find all possible solutions.

What is the first step when solving an absolute value equation that is not isolated?

The first step when solving an absolute value equation where the absolute value expression is not isolated is to isolate it on one side of the equation. For example, if you have $|x+1|+3=5$, subtract $3$ from both sides to get $|x+1|=2$. Once isolated, you can apply the rule to rewrite the absolute value equation as two separate equations without absolute values. This step is crucial because the solving method requires the absolute value to be alone on one side.