Problem 66 - Algebra for College Students

Question

Why is 2 the only solution for \(|x-2| \leq 0\) ?

Step-by-Step Solution

Answer

2 is the only solution because \(|x-2|\) is zero only when \(x = 2\).

Step 1: Understand Absolute Value Inequality

The absolute value \(|x-2|\) represents the distance between the number \(x\) and 2 on the number line. Since distance cannot be negative, \(|x-2|\) is always greater than or equal to zero.

Step 2: Analyze the Special Case of the Inequality

The inequality \(|x-2| \leq 0\) requires the expression inside the absolute value to be exactly zero. This is because absolute value cannot be negative, so it can only be less than or equal to zero if it equals zero.

Step 3: Solve the Equation Inside the Absolute Value

To satisfy \(|x-2| = 0\), solve the equation \(x - 2 = 0\). This simplifies to \(x = 2\).

Step 4: Conclusion Based on the Inequality

Since absolute values cannot be less than zero and can only equal zero at \(x = 2\), 2 is the only solution to the inequality \(|x-2| \leq 0\).

Key Concepts

Distance on Number Line

When it comes to understanding absolute value inequalities like \(|x-2| \), it helps to imagine the number line. \(|x-2| \) represents the distance between any number, \(x\), and the number 2 on this line. Think of absolute value as always measuring distance. Distance, by definition, cannot be negative.
For \(|x-2| = 0\), the distance between \(x\) and 2 is zero, meaning they are at the same point. That's why the equation \(|x-2| = 0\) solely results in \(x = 2\). No other numbers can reduce the distance to zero because every other number would be further away from 2.
Visualizing these expressions on a number line can greatly aid your understanding of why certain solutions are possible.

Solving Equations

Solving equations involving absolute values may seem tricky at first, but they follow a logical process. In the case of \(|x-2| \leq 0\), we have a special situation.
Normally, when solving \(|x-2| = c\), where \(c\) is a non-negative number, you consider two scenarios. \[\begin{align*} x - 2 &= c \ x - 2 &= -c \end{align*}\]
This allows \(x\) to take on values that are both \(c\) units away from 2 in either direction. But because zero is the smallest non-negative real number, the equation simply reduces to finding when \((x - 2) = 0\). Solving these equations leads to directly solving \(x - 2 = 0\), which simplifies to \(x = 2\).
The notion of distance via absolute value means any solution must satisfy the equation exactly when set equal to zero.

Absolute Value Properties

Absolute value has some key properties that are crucial to solving problems like these. One essential property is that the absolute value of a number is always non-negative. This means \(|x-2| \geq 0\) for all values of \(x\).
To satisfy \(|x-2| \leq 0\), both sides of the inequality must be exactly zero. Any deviation from zero would make the inequality false.
In general, the absolute value function converts any number to its distance from zero on a number line. It never dips below zero because you can't have a negative distance.

• \(|a| \geq 0\) for any real number \(a\)
• \(|a| = a\) if \(a\geq0\), and \(|a| = -a\) if \(a < 0\)
• \(|a - b|\) represents the distance between \(a\) and \(b\) on the number line

The properties of absolute value are integral because they ensure that its application in inequalities and equations remains consistent and predictable.