91Ó°ÊÓ

An inequality solution finds the range of values that satisfy a given inequality. To solve an inequality like the one in the exercise - \( 0 < x \leq 3 \), you need to understand what the inequality symbols mean. This inequality splits into two parts: \( x > 0 \) and \( x \leq 3 \).
\[ x > 0 \]
This says that x must be greater than zero. Basically, x can be any positive number.
\[ x \leq 3 \]
This says that x must be less than or equal to 3. So, x can be any number including 3 and those smaller than it.
When you put these together, you get a range of values for x: it must be greater than 0 and less than or equal to 3. So, to find a pair of inequalities that match \( 0 < x \leq 3 \), look for options describing this exact range.

Compound inequalities consist of two separate inequalities joined by the words 'and' or 'or'. In the exercise, the inequality \( 0 < x \leq 3 \) is a compound inequality. The use of 'and' indicates that both conditions must be satisfied at once. This is usually written as one continuous inequality, but it means both parts must be true. For example:
\( 0 < x \leq 3 \)
This can be split into:
\( x > 0 \ and \ x \leq 3 \)
Both conditions are necessary. If x only satisfies one part but not the other, it won't be part of the solution set. In compound inequalities using 'or', only one condition must be true. In this exercise, each option pairs two inequalities. Our task: identify the pair that correctly describes \( 0 < x \leq 3 \).

Understanding inequality symbols is essential for solving and interpreting inequalities. The main symbols include:

• \( < \) - 'less than'
• \( \leq \) - 'less than or equal to'
• \( > \) - 'greater than'
• \( \geq \) - 'greater than or equal to'

In \( 0 < x \leq 3 \), the '<' symbol means x cannot be zero, but must be any number larger than 0. The \( \leq \) symbol means that x can be any number up to and including 3. Knowing each symbol's meaning ensures accurate interpretation of the inequality range. In comparing the provided options, it's clear only Option (c) correctly aligns with \( 0 < x \leq 3 \). This option uses '>' and \( \leq \) precisely.