Problem 48 - Intermediate Algebra

Question

Solve each equation. See Example 5. $$ |4 x-7|=|3 x-21| $$

Step-by-Step Solution

Answer

The solutions are \(x = -14\) and \(x = 4\).

Step 1: Understand the Absolute Value Property

When solving an equation with absolute values like \(|a| = |b|\), it implies two separate cases: 1. \(a = b\)2. \(a = -b\). This property states that the absolute values are equal if either the values inside the absolute signs are equal, or if one is the negative of the other.

Step 2: Set Up the First Equation

For the equation \(|4x-7| = |3x-21|\), start by setting the expressions equal to each other:\[4x - 7 = 3x - 21\]

Step 3: Solve the First Equation

To solve \(4x - 7 = 3x - 21\), first eliminate \(3x\) from the right side by subtracting \(3x\) from both sides:\[4x - 3x - 7 = -21\]Simplify to get:\[x - 7 = -21\]Add 7 to both sides to solve for \(x\):\[x = -14\]

Step 4: Set Up the Second Equation

The second equation for the absolute values is formed by setting the expressions as opposites:\[4x - 7 = -(3x - 21)\]

Step 5: Simplify and Solve the Second Equation

Distribute the negative sign in the second equation:\[4x - 7 = -3x + 21\]Add \(3x\) to both sides:\[4x + 3x - 7 = 21\]Combine like terms:\[7x - 7 = 21\]Add 7 to both sides to isolate \(7x\):\[7x = 28\]Finally, divide by 7 to solve for \(x\):\[x = 4\]

Step 6: Verify Solutions

We found two potential solutions: \(x = -14\) and \(x = 4\). Verify by plugging them back into the original equation:1. For \(x = -14\): - LHS: \(|4(-14) - 7| = |-63| = 63\) - RHS: \(|3(-14) - 21| = |-63| = 63\)2. For \(x = 4\): - LHS: \(|4(4) - 7| = |9| = 9\) - RHS: \(|3(4) - 21| = |-9| = 9\) Both solutions satisfy the original equation.

Key Concepts

Properties of Absolute Values

Absolute value is a fundamental concept in mathematics that describes the distance of a number from zero on the number line, regardless of direction. For any real number \(x\), the absolute value is denoted as \(|x|\).
It is important to note that absolute values are always non-negative. Some key properties of absolute values are:

• \(|x| = x\) if \(x \geq 0\)
• \(|x| = -x\) if \(x < 0\)
• \(|-x| = |x|\)
• \(|x \, y| = |x| \cdot |y|\)

These properties are vital when solving equations involving absolute values, as they simplify and break down complex scenarios into more manageable ones.

Solving Equations with Absolute Values

To solve absolute value equations, we rely on the fundamental principle that if \(|a| = |b|\), then either \(a = b\) or \(a = -b\).
This allows us to explore two scenarios for each equation:

• The expressions inside the absolute values are equal.
• The expressions are negatives of each other.

Consider the equation \(|4x-7| = |3x-21|\). This equation can be separated into two cases:

• \(4x-7 = 3x-21\)
• \(4x-7 = -(3x-21)\)

By tackling these cases individually, we can find all possible solutions to the original equation.

Algebraic Equations

An algebraic equation is any equation formed by polynomial expressions and can involve constants, variables, or combinations thereof.
Solving these equations typically involves applying algebraic properties such as addition, subtraction, multiplication, division, and distribution.

• Identify and isolate terms with variables.
• Combine like terms.
• Use inverse operations to solve for the variable.

In our exercise, first address the equation \(4x-7 = 3x-21\).
By isolating \(x\) using algebraic techniques (subtracting \(3x\), adding 7, etc.), we solve for possible values of \(x\).
Repeat these steps for the alternative scenario \(4x-7 = -(3x-21)\).

Verifying Solutions

Once potential solutions are obtained from an absolute value equation, verification is crucial. This is to ensure these solutions work for the original equation.
We substitute each value back into the equation to check its validity.

• For \(x = -14\), verify if \(|4(-14) - 7|\) equals \(|3(-14) - 21|\).
• For \(x = 4\), check if \(|4(4) - 7|\) equals \(|3(4) - 21|\).

Both checks should result in equal values on either side of the equation, confirming the solutions are correct.

Intermediate Algebra Concepts

Intermediate algebra bridges basic algebra and more advanced mathematical concepts, providing strategies to tackle complex equations and inequalities.
Understanding absolute value as both a concept and a tool is part of intermediate algebra.

• Leverage the properties of equality to rearrange and simplify equations.
• Recognize patterns such as linear and quadratic expressions.
• Adjust equations using inverse operations effectively.

By mastering intermediate algebra methods, equations with absolute values, like the given exercise, become more approachable and solvable.