91Ó°ÊÓ

Inequalities are similar to equations, but instead of an equal sign, they use symbols to show less than, greater than, less than or equal to, or greater than or equal to. The objective of solving inequalities is to find all possible values of a variable that make the inequality true. It's a fundamental process that requires performing operations similar to those used in solving equations, such as addition, subtraction, multiplication, and division.

In the given exercise, we have a compound inequality composed of two separate inequalities:

• \( \frac{2}{3}x - \frac{5}{6} \leq 0 \)
• \( -3x < -2 \)

**Step-by-step solutions for these inequalities** involve isolating the variable on one side of the inequality.
- For the first inequality, adding \( \frac{5}{6} \) to both sides simplifies the problem, then multiplying both sides by the reciprocal \( \frac{3}{2} \) leads to the solution \( x \leq \frac{5}{4} \).- In the second inequality, dividing by \(-3\) (and remembering to flip the inequality sign) gives us \( x > \frac{2}{3} \).

Make sure to always reverse the inequality symbol when multiplying or dividing both sides by a negative number, which is a unique rule when handling inequalities.

After finding the solutions to the inequalities, the next step is graphing these solutions to visualize the range of values that satisfy the inequalities. Graphing inequalities on a number line helps to see where the solutions overlap or intersect, especially for compound inequalities. This can support better understanding about the inequalities' relationships and provide a clear visual of the solution set.

In our exercise, we need to graph the compound solution \( \frac{2}{3} < x \leq \frac{5}{4} \):

• Place an open circle on the number line at \( \frac{2}{3} \). The open circle indicates that \( x \) is not equal to \( \frac{2}{3} \), only greater than it.
• Draw a closed circle at \( \frac{5}{4} \). The closed circle shows that \( x \) can be equal to \( \frac{5}{4} \).
• Shade the region on the number line between \( \frac{2}{3} \) and \( \frac{5}{4} \) to represent all possible values of \( x \) within these boundaries.

This graph visually simplifies the understanding of which values \( x \) can take to satisfy the compound inequality.

A solution set is the collection of all possible solutions that satisfy an inequality or a system of inequalities. Understanding solution sets is crucial because they tell us what values the variable can take.

In compound inequalities, like the one in our exercise, the solution set is derived from the overlap between the solutions of the individual inequalities. It's important to find this intersection because it represents the values that make both inequalities true simultaneously.

• For the inequality \( x \leq \frac{5}{4} \), our solution set includes any value up to and including \( \frac{5}{4} \).
• For \( x > \frac{2}{3} \), the values start from just above \( \frac{2}{3} \) and extend upwards.

The solution set of the compound inequality \( \frac{2}{3} < x \leq \frac{5}{4} \) is the range where these two conditions overlap.
This means that all values of \( x \) are valid starting just after \( \frac{2}{3} \) up to \( \frac{5}{4} \), including \( \frac{5}{4} \) itself, and we exclude exactly \( \frac{2}{3} \). This overlap results in a defined, clear range for \( x \).