Problem 26 - Algebra for College Students

Question

Solve each equation. $$ \left|2-\frac{5}{2} x\right|=14 $$

Step-by-Step Solution

Answer

The solutions are \( x = -\frac{24}{5} \) and \( x = \frac{32}{5} \).

Step 1: Set up the equation

The given equation is \ \( \left|2-\frac{5}{2} x\right|=14 \). Since it's an absolute value equation, it splits into two separate linear equations to solve. These are: \[ 2-\frac{5}{2}x = 14 \ \text{and} \ 2-\frac{5}{2}x = -14 \]

Step 2: Solve the first equation

Solve \ \( 2-\frac{5}{2}x = 14 \) by isolating \(x\). \ \begin{align*} 2-\frac{5}{2}x & = 14 \ -\frac{5}{2}x & = 12 \ x & = -\frac{24}{5} \end{align*} \ Therefore, \( x = -\frac{24}{5} \).

Step 3: Solve the second equation

Solve \ \( 2-\frac{5}{2}x = -14 \) by isolating \(x\). \ \begin{align*} 2-\frac{5}{2}x & = -14 \ -\frac{5}{2}x & = -16 \ x & = \frac{32}{5} \end{align*} \ Therefore, \( x = \frac{32}{5} \).

Step 4: Combine solutions

The solutions to the equation \( \left|2-\frac{5}{2} x\right|=14 \) are the two values obtained: \( x = -\frac{24}{5} \) and \( x = \frac{32}{5} \).

Key Concepts

linear equations

Linear equations are a type of equation that forms a straight line when graphed.
They typically have the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.
In the given exercise, solving the absolute value equation splits into two linear equations.
Consider these two separate cases:

• \[2 - \frac{5}{2}x = 14\]
• \[2 - \frac{5}{2}x = -14\]

Both of these are linear equations. When we solve them, we find the values for \(x\) that make the equation true.

Simplifying linear equations generally involves:

• Combining like terms.
• Using addition or subtraction to move terms containing the variable to one side of the equation.
• Using multiplication or division to isolate the variable.

isolation of variables

Isolation of variables is a key step in solving equations.
The goal is to get the variable on one side of the equation by itself.
This typically involves a few steps:

• First, move any constant terms away from the side with the variable by adding or subtracting them.
• Then, divide or multiply to get the variable by itself.

In our given equation, \[2 - \frac{5}{2}x = 14\], we first subtract 2 from both sides to get:
\[ -\frac{5}{2}x = 12 \]
Then, multiply both sides by the reciprocal of \[-\frac{5}{2}\], which is \[-\frac{2}{5}\], to isolate \(x\):
\[ x = -\frac{24}{5} \]
For the second equation, \[ 2 - \frac{5}{2}x = -14 \], we follow similar steps:
First, subtract 2 from both sides:
\[ -\frac{5}{2}x = -16 \]
Multiply by the reciprocal of \[-\frac{5}{2} \] to isolate \(x\):
\[ x = \frac{32}{5} \]
By isolating the variable, we can determine the specific values for \(x\) that satisfy each equation.

absolute value properties

Absolute value refers to the distance a number is from zero on the number line.
It is always non-negative because distance cannot be negative.
The absolute value of a number \(a\) is written as \(|a|\).
For example, \( |-7| = 7 \) and \( |4| = 4 \).

In absolute value equations, \( |expression| = k \), the expression inside the absolute value can be equal to \(k\) or \(-k\).
This means we create two separate linear equations to solve:

• \[ expression = k \]
• \[ expression = -k \]

For instance, \(|2 - \frac{5}{2}x| = 14\) leads to two cases:

• \[2 - \frac{5}{2}x = 14\]
• \[2 - \frac{5}{2}x = -14\]

By solving these, we find the possible values for \(x\) that satisfy the original absolute value equation.
This property is crucial to approach and solve absolute value problems systematically.