91Ó°ÊÓ

Absolute value is a fundamental concept in mathematics. It represents the non-negative magnitude of a real number, without regard to its sign. For example, the absolute value of both 3 and -3 is 3, denoted as \(|3| = 3\) and \(|-3| = 3\). This concept is vital when solving absolute value equations, as it allows us to consider both positive and negative solutions.

In the given exercise, we deal with an equation of the form \(|x| = c\), where \(c\) is a positive constant. The absolute value equation \(|x| = c\) means that \(x\) can be either \(+c\) or \(-c\). This dual nature of absolute values is crucial when solving such equations, as it ensures we capture all possible solutions. Always remember the following:

• Absolute value represents distance from zero on the number line.
• \(|a| = b\) implies \(a = b\) or \(a = -b\).

In mathematics, real solutions refer to the set of solutions to an equation that are real numbers. Real numbers include all the numbers on the number line—both positive and negative numbers, as well as zero.

Considering the exercise, the goal was to find the real solutions of the equation \(\frac{3}{4}|x| = 9\). Following the steps:

First, isolate the absolute value term:
\(\frac{3}{4}|x| = 9\) becomes \(|x| = 12\).
This indicates that \(x\) can be either 12 or -12.

Thus, the real solutions to the equation are:

• \(x = 12\)
• \(x = -12\)

We conclude that these solutions are real numbers and can be represented on the number line.

Equations are mathematical statements asserting that two expressions are equal, shown using the equals sign (=). When dealing with absolute value equations, it’s essential to follow the correct steps to isolate and solve the equation accurately.

Consider the equation provided: \[ \frac{3}{4}|x| = 9 \]

Steps to solve it:

• Isolate the absolute value by multiplying both sides by the reciprocal of \( \frac{3}{4} \):
  \[ \frac{4}{3} \times \frac{3}{4}|x| = 9 \times \frac{4}{3} \]
  This simplifies to \[|x| = 12 \].
• Once we have \[|x| = 12 \], recognize that the absolute value equation implies two scenarios:
  \[ x = 12 \] and \[ x = -12 \].

Therefore, the real solutions to the equation \(\frac{3}{4}|x| = 9\) are \(x = 12\) and \(x = -12\).

Always remember, isolating the absolute value term is key to solving these types of equations correctly.