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Magazine Chapter 2: Problem 5
Solve each absolute value equation. See Examples 1 through 7. $$ |2 x-5|=9 $$
Short Answer
Expert verified
The solutions are \(x = 7\) and \(x = -2\).
Step by step solution
01
Understand the Absolute Value Definition
The absolute value \(|x|\) represents the distance of \(x\) from 0 on the number line. Therefore, \(|2x - 5| = 9\) can be interpreted as \(2x - 5\) being 9 units away from 0. This creates two possible equations.
02
Set Up Two Equations
Because the absolute value represents a distance that can be in either direction of zero, we set up two separate equations:1. \(2x - 5 = 9\)2. \(2x - 5 = -9\).
03
Solve the First Equation
Solve \(2x - 5 = 9\):- Add 5 to both sides: \(2x - 5 + 5 = 9 + 5\) results in \(2x = 14\).- Divide by 2: \(x = \frac{14}{2}\) which simplifies to \(x = 7\).
04
Solve the Second Equation
Solve \(2x - 5 = -9\):- Add 5 to both sides: \(2x - 5 + 5 = -9 + 5\) results in \(2x = -4\).- Divide by 2: \(x = \frac{-4}{2}\) which simplifies to \(x = -2\).
05
Verify Solutions
Plug both solutions back into the original absolute value equation to verify.- For \(x = 7\): \(|2(7) - 5| = |14 - 5| = |9| = 9\) which is correct.- For \(x = -2\): \(|2(-2) - 5| = |-4 - 5| = |-9| = 9\) which is also correct. 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.
Solving Equations
Solving equations is a fundamental skill in algebra, where the goal is to find the value of the unknown variable that makes the equation true. Each equation operates as a balance, with the left and right sides needing to be equal.
To solve this, perform basic arithmetic operations:
- Identify the type of equation you are working with. In this case, it's an absolute value equation.
- Your task is to isolate the variable, which often involves inverse operations.
To solve this, perform basic arithmetic operations:
- Add 5 to both sides to remove the constant term: \(2x - 5 + 5 = 9 + 5\).
- Then, divide both sides by 2 to solve for \(x\): \(x = \frac{14}{2}\).
Absolute Value
Absolute value is a concept that indicates the "distance" of a number from zero on the number line, ignoring whether it is positive or negative. The symbol for absolute value involves placing vertical bars around a number or expression, like \(|x|\).
For an equation such as \(|2x - 5| = 9\), you're looking at two potential scenarios where the expression inside the absolute value could result in positive or negative 9.
For an equation such as \(|2x - 5| = 9\), you're looking at two potential scenarios where the expression inside the absolute value could result in positive or negative 9.
- This is due to the symmetrical nature of distance on a number line: \(2x - 5\) could equal 9 or -9.
- By setting up each case, \(2x - 5 = 9\) and \(2x - 5 = -9\), you encompass all possibilities.
Intermediate Algebra
Intermediate algebra is a branch of mathematics that builds upon basic algebraic principles, introducing more complex concepts and types of equations, like absolute value equations. In this particular exercise, we tackled an intermediate-level problem that involves understanding and manipulating absolute value expressions.
In approaching these problems:
Developing proficiency in these areas serves as a stepping stone to tackling even more advanced mathematical topics.
In approaching these problems:
- First, grasp the problem's context and define all relevant expressions and operations.
- Use skills like setting up multiple equations from a single absolute value equation and verify solutions by back-substitution.
Developing proficiency in these areas serves as a stepping stone to tackling even more advanced mathematical topics.
