Problem 129 - College Algebra

Question

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Find \(b\) such that \(\frac{7 x+4}{b}+13-x\) has a solution set given by \(|-6|\)

Step-by-Step Solution

Answer

The value of 'b' that makes the equation \( \frac{7x + 4}{b} + 13 - x = |-6|\) true is \(\frac{7x + 4}{x + 13}\).

Step 1: Determine the Value of the Absolute Expression

The absolute value of -6 is 6.

Step 2: Substitute the value in the equation

In the equation \(\frac{7x + 4}{b} + 13 - x = |-6|\), replace |-6| with 6. This results in the equation \(\frac{7x + 4}{b} + 13 - x = 6\)

Step 3: Simplify the Equation

Now, the aim is to isolate 'b'. First let's bring like terms together. This renders the equation as \(\frac{7x + 4}{b} + (13 - x) = 6\). Simplify \(13 - x\) to its simplest form, the equation now becomes \(\frac{7x + 4}{b} + 13 - x = 6\) which can be further simplified to \(\frac{7x + 4 - b(13 - x)}{b} = 6\)

Step 4: Cross-multiply to Solve for 'b'

Now, cross-multiply to get rid of the fraction: \(7x + 4 - b(13 - x) = 6b\). Distribute '-b' across '(13 - x)' to get: \(7x + 4 - 13b + bx = 6b\). Rearrange the equation to isolate 'b' on one side of the equation: \(b(x + 13) = 7x + 4\).

Step 5: Solve for 'b'

Finally, by dividing each side by \((x + 13)\) will give us the value of b = \(\frac{7x + 4}{x + 13}\).

Key Concepts

Absolute Value

Absolute value is a fundamental concept in mathematics, representing the distance of a number from zero, regardless of direction on the number line.
When we take the absolute value of any number, we are essentially finding its non-negative counterpart. For example, the absolute value of -6 is 6 because -6 is six units away from zero, as is +6.

• Notation: The absolute value of a number is denoted by two vertical bars, like this: \(|x|\).
• The absolute value of a positive number is the number itself.
• The absolute value of a negative number is its positive counterpart.

Understanding absolute values helps simplify equations, especially when dealing with inequalities and other algebraic expressions.

Equation Simplification

Equation simplification is the process of reducing an algebraic equation to its simplest or most manageable form. The goal is to make the equation easier to solve by eliminating unnecessary terms and combining like terms.
In the given exercise, simplification involves substituting the absolute value and rearranging terms.

• Combine like terms: Group similar variables and constants together.
• Simplify expressions: Break down complex parts into simpler units.

These steps provide clarity and often lead to the quick isolation of variables, preparing the equation for further solving steps.

Cross-Multiplication

Cross-multiplication is a powerful technique used primarily to solve algebraic equations involving fractions. It involves multiplying across the equals sign in a way that eliminates fractions, leaving a simpler equation to work with.
To perform cross-multiplication correctly:

• Multiply the numerator of one side by the denominator of the other side.
• Set the products equal to solve for the variable.

This technique is beneficial when dealing with equations that involve terms separated by fractions. In the provided solution, cross-multiplication helps eliminate the fraction and simplifies the path towards solving for the variable 'b'.

Isolation of Variables

Isolation of variables is a crucial step in solving equations, where the goal is to get the variable of interest by itself on one side of the equation. This allows for easy calculation of the variable's value.
Here are the basic steps to isolate a variable like 'b':

• Perform inverse operations to move terms involving the variable to one side of the equation.
• Factor out the variable if necessary, to ensure it is isolated.
• Divide by any remaining coefficients to fully isolate the variable.

In the solution, 'b' is isolated by rearranging terms, factoring, and finally dividing, enabling us to express 'b' solely in terms of other known quantities.