Problem 18 - Algebra for College Students

Question

Solve each equation. $$ |-6 x+5|=4 $$

Step-by-Step Solution

Answer

x = \frac{1}{6} \text{ or } x = \frac{3}{2}.

Step 1 - Understand Absolute Value Properties

Absolute value equations require us to consider both the positive and negative scenarios of the expression inside the absolute value. So, we need to solve for both \( -6x + 5 = 4 \) and \( -6x + 5 = -4 \).

Step 2 - Solve the First Equation

Let's solve \( -6x + 5 = 4 \). Subtract 5 from both sides: \( -6x + 5 - 5 = 4 - 5 \), which simplifies to \( -6x = -1 \). Next, divide both sides by -6: \( x = \frac{1}{6} \).

Step 3 - Solve the Second Equation

Now solve \( -6x + 5 = -4 \). Subtract 5 from both sides: \( -6x + 5 - 5 = -4 - 5 \), which simplifies to \( -6x = -9 \). Next, divide both sides by -6: \( x = \frac{3}{2} \).

Key Concepts

absolute value properties

Absolute value refers to the distance a number is from zero on a number line. This distance is always a non-negative value. For example, both 3 and -3 have an absolute value of 3.
When solving absolute value equations, like \(|−6x+5|=4\), we need to consider both the positive and negative scenarios. This is because the expression inside the absolute value, \(-6x + 5\), must equal both 4 and -4 to generate possible x values.
Therefore, we split the original absolute value equation into two separate linear equations:

• \(-6x + 5 = 4\)
• \(-6x + 5 = -4\)

Solving these separate equations will give us all potential solutions to the original problem.

linear equations

Linear equations are mathematical statements of equality involving variables whose powers are all equal to one. An equation like \(-6x + 5 = 4\) is considered linear.
The main property of linear equations is that they graph as straight lines. When dealing with absolute value questions, solving the split linear equations helps us identify where these lines intersect or meet certain conditions, like our distance from zero.
For the equation \(-6x + 5 = 4\), we move forward by turning it into a more straightforward form by isolating the variable x. This involves reversing arithmetic operations—subtracting or adding numbers and then dividing or multiplying them—to solve for x.

equation solving steps

Here are the steps to solve equations, particularly focusing on our example: \(|-6x+5|=4\):

• **Step 1:** Recognize absolute value properties. Split the equation into two linear equations: \(-6x + 5 = 4\) and \(-6x + 5 = -4\).
• **Step 2:** Solve the first equation \(-6x + 5 = 4\).
  Subtract 5 from both sides:\br> \(-6x + 5 - 5 = 4 - 5\)
  This simplifies to \(-6x = -1\).
  Divide by -6:
  \(x = \frac{1}{6} \)
• **Step 3:** Solve the second equation \(-6x + 5 = -4\).
  Subtract 5 from both sides:
  \(-6x + 5 - 5 = -4 - 5\)
  This simplifies to \(-6x = -9 \).
  Divide by -6:
  \(x = \frac{3}{2} \)

By following these steps, you will find all possible solutions to \(|-6x+5|=4\).