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Magazine Chapter 4: Problem 48
Solve each equation. See Example 5. $$ |4 x-7|=|3 x-21| $$
Short Answer
Expert verified
The solutions are \(x = -14\) and \(x = 4\).
Step by step solution
01
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.
02
Set Up the First Equation
For the equation \(|4x-7| = |3x-21|\), start by setting the expressions equal to each other:\[4x - 7 = 3x - 21\]
03
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 \(x\):\[x = -14\]
04
Set Up the Second Equation
The second equation for the absolute values is formed by setting the expressions as opposites:\[4x - 7 = -(3x - 21)\]
05
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 \(x\):\[x = 4\]
06
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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 \(x < 0\)
- \(|-x| = |x|\)
- \(|x \, y| = |x| \cdot |y|\)
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:
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.
- \(4x-7 = 3x-21\)
- \(4x-7 = -(3x-21)\)
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.
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 \(4x-7 = -(3x-21)\).
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.
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.
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 \(|3(4) - 21|\).
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.
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.
