Given that S and T are sets of real numbers such that S = {x : 0 $\leq$ x $\leq$ 5} and T = {x : − 2 < x < 3} Find S $\cup$ T

1. Analyze the Question

We are given two sets of real numbers:

• Set S: $\{x : 0 \leq x \leq 5\}$ (This includes all numbers from 0 to 5, including both 0 and 5).
• Set T: $\{x : -2 < x < 3\}$ (This includes all numbers between -2 and 3, excluding -2 and 3).
• Goal: Find the Union ($S \cup T$). The union of two sets contains all elements that are in either S, or T, or both.

2. Explain the Concept

• Union ($\cup$): Think of this as "combining" the two sets. You look for the lowest possible value covered by either set and the highest possible value covered by either set.
• Interval Notation: • $S = [0, 5]$ • $T = (-2, 3)$
• When you combine these on a number line, the range starts from the lowest value in $T$ (which is $-2$) and ends at the highest value in $S$ (which is $5$).

3. Show the Solution

• Step 1: Visualize the number line. • Set $T$ starts just after $-2$ and goes up to $3$. • Set $S$ starts at $0$ and goes up to $5$.
• Step 2: Combine the ranges. • The combined range starts at $-2$. Since $-2$ is not included in $T$ (it is a "less than" sign), it remains excluded: $-2 < x$. • The combined range ends at $5$. Since $5$ is included in $S$ (it is a "less than or equal to" sign), it remains included: $x \leq 5$.
• Step 3: Write the union. • Combining these, we get: $-2 < x \leq 5$.

4. Identify the Answer

Comparing our result to the given options: A. $-3 < x \leq 3$ B. $-2 < x \leq 5$ C. $2 < x \geq -5$ D. $-1 < x \geq \leq 2$

The correct option is B.

5. Exam Tip

When dealing with inequalities, always draw a quick sketch of the number line. Use an open circle ($\circ$) for "less than" ($<$) or "greater than" ($>$), and a shaded circle ($\bullet$) for "less than or equal to" ($\leq$) or "greater than or equal to" ($\geq$). This makes it much easier to see where the combined set begins and ends!