Problem 24 - College Algebra with Modeling and Visualization

Question

Solve the absolute value equation. $$|2-3 x|=1$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{3}\) and \(x = 1\).

Step 1: Understand the Absolute Value Expression

The absolute value of a number represents its distance from zero on the number line, regardless of direction. The equation \(|2-3x|=1\) indicates that \(2-3x\) can be either 1 unit away or -1 unit away from zero. This creates two possible scenarios for \(2-3x\).

Step 2: Set Up the Two Possible Scenarios

The absolute value equation \(|2-3x|=1\) implies two equations: \(2-3x=1\) (when the expression within the absolute value is positive or zero) and \(2-3x=-1\) (when the expression is negative). We will solve for \(x\) in both cases to find all solutions to the equation.

Step 3: Solve the First Scenario

For the first equation, \(2-3x=1\):Subtract 2 from both sides:\[-3x = 1 - 2\]\[-3x = -1\]Divide both sides by -3:\[x = \frac{-1}{-3}\]\[x = \frac{1}{3}\]

Step 4: Solve the Second Scenario

For the second equation, \(2-3x=-1\):Subtract 2 from both sides:\[-3x = -1 - 2\]\[-3x = -3\]Divide both sides by -3:\[x = \frac{-3}{-3}\]\[x = 1\]

Step 5: Solutions Verification

We have two potential solutions: \(x = \frac{1}{3}\) and \(x = 1\). Substitute each back into the original equation to verify:For \(x = \frac{1}{3}\):\[|2 - 3(\frac{1}{3})| = |2 - 1| = |1| = 1\] which is true.For \(x = 1\):\[|2 - 3(1)| = |2 - 3| = |-1| = 1\] which is also true.Both solutions satisfy the original equation.

Key Concepts

Solving Equations

An equation is like a mathematical statement saying two expressions are equal. When we talk about solving equations, especially absolute value equations, it means finding what values satisfy the equation, or make it true. Let's break down the process with the example equation \(|2-3x|=1\). Here, solving involves figuring out the values of \(x\) that make the absolute value expression equal to 1.

To start, remember that the absolute value equation \(|a|=b\) suggests two scenarios: \(a = b\) and \(a = -b\). This means the expression inside the absolute value can either be positive or negative. For our case:

• Scenario 1: \(2-3x = 1\)
• Scenario 2: \(2-3x = -1\)

For each scenario, we rearrange the equation to solve for \(x\). After simplifying:

• From \(2-3x = 1\), we get \(x = \frac{1}{3}\).
• From \(2-3x = -1\), we find \(x = 1\).

Checking these in the original equation confirms they both work, showing there are two solutions: \(x = \frac{1}{3}\) and \(x = 1\).

Algebraic Expressions

When dealing with equations like \(|2-3x| = 1\), understanding algebraic expressions is crucial. An algebraic expression combines numbers, variables, and arithmetic operations (like addition or multiplication) into a meaningful mathematical phrase. For the provided equation, \(2-3x\) is the algebraic expression inside the absolute value.

Let's break it down.

• "2" is a constant. It's a number by itself.
• "3x" means 3 times some number x. This is a linear term because it doesn't involve powers or roots of \(x\).

Together, these elements form a linear expression "2 - 3x". This expression is embedded within the absolute value, indicating its distance from zero, which affects the resulting scenarios when solving the equation.

Algebraic expressions are crucial because they represent the operations that shape the equation's solutions. Understanding how to manipulate them by adding, subtracting, or isolating variables is key to reaching a solution. By transforming \(2-3x = 1\) or \(2-3x = -1\), we isolate \(x\) and solve for it. This is a fundamental skill in algebra.

Distance on the Number Line

The concept of distance on the number line is vital in understanding absolute value equations. Absolute value measures how far a number is from zero, no matter the direction. Think of it as a distance that can't be negative.

For the equation \(|2-3x|=1\), what we're saying is that the expression \(2-3x\) is exactly 1 unit away from zero on the number line. This distance aspect has two possibilities: moving 1 unit to the right or 1 unit to the left, corresponding to 1 or -1.

Visualizing this, imagine a number line:

• Moving 1 unit right from zero lands us at position 1, so \(2-3x\) could equal 1.
• Moving 1 unit left places us at position -1, so \(2-3x\) might also equal -1.

The absolute value strips away the direction, focusing only on how far away we are, ensuring we measure non-negative distance. In this sense, absolute values act much like rulers on a number line, measuring distance without regard for direction, helping you find all possible solutions.