Problem 58 - Calculus with Concepts in Calculus

Question

Solve the equation. $$ |x-1|=|2 x+1| $$

Step-by-Step Solution

Answer

The solutions are \(x = -2\) and \(x = 0\).

Step 1: Understanding the Absolute Value Equation

We begin with the equation \(|x-1| = |2x+1|\). This tells us the absolute value of \(x-1\) is equal to the absolute value of \(2x+1\). Since absolute value represents the distance from zero, it can lead to two different equations where either both expressions inside the absolute values are equal or they are negatives of each other.

Step 2: Scenario 1: Equal Expressions

First, we assume that the expressions inside the absolute values are equal: \(x-1 = 2x+1\). Solve for \(x\): \(x - 1 = 2x + 1\)Subtract \(x\) from both sides: \(-1 = x + 1\)Subtract 1 from both sides: \(x = -2\).

Step 3: Scenario 2: Negative Expressions

Next, we consider that one expression could be the negative of the other: \(x-1 = -(2x+1)\). Solve for \(x\): \(x - 1 = -2x - 1\)Add \(2x\) to both sides: \(3x - 1 = -1\)Add 1 to both sides: \(3x = 0\)Divide by 3: \(x = 0\).

Step 4: Verify Solutions

Substitute \(x = -2\) and \(x = 0\) back into the original equation to check validity. For \(x = -2\):\(|-2-1| = |-4| = 3\) and \(|2(-2) + 1| = |-3| = 3\). This is correct.For \(x = 0\):\(|0-1| = |-1| = 1\) and \(|2(0) + 1| = |1| = 1\). This is also correct.Thus, both solutions satisfy the original equation.

Key Concepts

algebraic equations

Algebraic equations are mathematical expressions set equal to each other. They involve variables like \(x\) and coefficients (numbers multiplying the variables) and may include constants (fixed numbers). Algebra is used to represent real-world situations and solve problems involving unknown values.

Equations can be very simple, like \(x + 3 = 5\), or more complex with multiple terms and variables. Solving equations is fundamental in algebra, as it allows us to find the value of unknowns.

• An equation can be solved by applying different rules or operations evenly across its terms.
• The goal is to isolate the variable on one side of the equation to determine its value.

In equations with absolute values, we often face more than one scenario to solve since absolute value can represent two potential states (positive or negative). Hence, understanding absolute value behaviors is crucial when dealing with equations like \(|x-1| = |2x+1|\).

equation solving strategies

Equation solving strategies involve various steps and techniques to find the value of a variable that makes an equation true. Understanding these strategies is essential to solving not just simple, but also complex algebraic equations.

Here are a few strategies often used:

• Simplification: Combine like terms and simplify each side of the equation as much as possible.
• Isolation: Rearrange the equation to get the variable on one side and constants on the other. This often involves adding, subtracting, multiplying, or dividing both sides of the equation.
• Checking Solutions: Substitute the obtained values back into the original equation to ensure they satisfy it.

For absolute value equations specifically, it's important to consider that the expressions inside absolute values can either be equal or be negatives of each other. This leads to different scenarios that need to be checked individually, as shown in steps for solving \(|x-1| = |2x+1|\).

absolute value properties

Absolute value properties reflect the concept of taking the "distance" from zero. Whether the original value is positive or negative, the absolute value is always non-negative and represented as \(|x|\). Understanding this can help in solving absolute value equations.

When dealing with equations such as \(|x-1| = |2x+1|\), remember:

• Equal Expressions: The equation inside the absolute value can directly equal each other.
• Negative Counterparts: One expression could be the negative of the other, as an absolute value essentially removes negatives.

Each property reflects a different scenario or approach to consider. In other words, you must often evaluate these situations separately because absolute values allow the same distance to be represented by two different points (one positive, one negative) on the number line. This dual nature is why absolute value equations generally have more than one solution.