Problem 70 - College Algebra Essentials

Question

Solve each absolute value equation or indicate that the equation has no solution. $$ 7|3 x|+2=16 $$

Step-by-Step Solution

Answer

The solutions for the equation are \(x=2/3\) and \(x=-2/3\)

Step 1: Isolate the Absolute Value

In the original equation \(7|3 x|+2=16\), first subtract 2 from both sides of the equation, which yields \(7|3 x|=14\). Then divide by 7 on both sides to isolate the absolute value, which gives \(|3x|=2\)

Step 2: Split into Two Linear Equations

Once the absolute value is isolated, it's possible to translate this into two linear equations: \(3x=2\) and \(3x=-2\).

Step 3: Solve Both Equations

Solve these two equations separately. For the first equation, \(3x=2\), divide both sides by 3, which yields \(x=2/3\). For the second equation, \(3x=-2\), divide both sides by 3, which yields \(x=-2/3\).

Step 4: Verify the solutions

Replace \(x\) with \(2/3\) and \(-2/3\) in the original equation \(7|3x|+2=16\) to verify the solutions. Both satisfy the original equation as \(7|3*(2/3)|+2 = 16\) and \(7|3*(-2/3)|+2 = 16\), so both \(x=2/3\) and \(x=-2/3\) are valid solutions

Key Concepts

Isolate the Absolute Value

The initial step in solving an absolute value equation is to isolate the absolute value expression. For instance, starting with the equation \[7|3x| + 2 = 16,\]we first need to get rid of the constant term on the side of the equation with the absolute value. By simply subtracting 2 from both sides, we revise the equation to \[7|3x| = 14.\] Next, divide each side by 7 to ensure that the absolute value is completely alone, resulting in \[|3x| = 2.\]By isolating the absolute value, we make it easier to handle the expression inside, which in turn simplifies the problem. This strategy of isolation is key; always start by removing any additional numbers or coefficients that directly affect or multiply the absolute value term. This foundational step prepares us for the next processes in solving the equation.

Split into Linear Equations

Once the absolute value is isolated, the equation \[|3x| = 2\]can be understood as having two potential cases. This dual nature is embedded in the definition of absolute values, which accounts for both positive and negative possibilities of what's contained inside. Thus, the equation splits into two separate linear equations:

• \(3x = 2\)
• \(3x = -2\)

To determine the possible values for \(x\), we must solve these equations separately. These equations reflect the two scenarios where the expression inside the absolute value could equal a positive or negative value that yields the original absolute value expression. This splitting process essentially breaks down a single absolute value equation into two simpler problems.

Verify Solutions

After solving the linear equations, we obtain potential solutions for \(x\) which need confirmation. Replacing these values back into the original equation \[7|3x| + 2 = 16\]ensures that our solutions are correct and satisfy the equation completely.Let's test the values:

• Substitute \(x = \frac{2}{3}\) into the equation: \[7|3 \times \frac{2}{3}| + 2 = 16\] simplifies to \[7 \times 2 + 2 = 16,\] which is accurate.
• Substitute \(x = -\frac{2}{3}\) into the original equation: \[7|3 \times -\frac{2}{3}| + 2 = 16\] also simplifies to \[7 \times 2 + 2 = 16,\] which confirms the solution as correct.

Verifying solutions is essential to ensure that no errors were made in earlier steps and that both values truly solve the equation. This step provides confidence in the correctness of our findings by affirming that initial equation conditions are met precisely.