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Interval notation is a way to describe sets of numbers, where the intervals denote the range of values that satisfy certain conditions. For instance, when we say \( p < 32 \), it means that every number less than 32 is a part of the solution. We represent this in interval notation as \( (-\infty, 32) \). The round brackets, also called parentheses, indicate that the endpoints are not included.
To better understand:

• Use parentheses \( ( \) and \( ) \) for values that are NOT included in the interval.
• Use square brackets \[ \) and \( \] for values that ARE included in the interval.

In our case, because 32 is not included, we use a parenthesis. The symbol \( -\infty \) means continuing infinitely in the negative direction without bound. This is always paired with a parenthesis.

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions using various rules and properties. Here’s a step-by-step guide using the given problem:
1. Distribute the fractions: Multiply \( \frac{2}{3} \) and \( \frac{5}{6} \) through the parentheses:
\[ \frac{2}{3}(p + 3) = \frac{2}{3}p + 2 \]
\[ \frac{5}{6}(p - 4) = \frac{5}{6}p - \frac{10}{3} \]
2. Eliminate fractions: Multiply every term by 6 (LCM of 3 and 6):
\[ 6 \times \frac{2}{3}p + 6 \times 2 > 6 \times \frac{5}{6}p - 6 \times \frac{10}{3} \] This simplifies to:
\[ 4p + 12 > 5p - 20 \]
3. Isolate the variable: Move all terms involving \( p \) to one side and constants to the other:
Subtract \( 5p \) from both sides:
\[ 4p + 12 - 5p > -20 \]
which simplifies to: \[ -p + 12 > -20 \]
Subtract 12 from both sides:
\[ -p > -32 \]
Multiply both sides by \( -1 \) (remember to reverse the inequality):
\[ p < 32 \]

Graphing inequalities shows a visual representation of their solution sets. When we graph the inequality \( p < 32 \) on a number line:

• Draw a number line.
• Mark the point 32 with an open circle to indicate it is not included.
• Shade the region to the left of 32, showing all values less than 32.

This visual helps you immediately see the range of values that satisfy the inequality.
Remember:

• An open circle means the number is not included (e.g., less than or greater than).
• A closed circle means the number is included (e.g., less than or equal to, or greater than or equal to).

Graphing makes it easier to understand and confirm the solution set visually. It’s a great way to double-check your algebraic solution.