91Ó°ÊÓ

Understanding linear inequalities is crucial for solving many kinds of mathematical problems. In essence, a linear inequality is similar to a linear equation, but instead of finding a precise answer, it involves a range of solutions. For instance, the inequality from the exercise,
\( y \geq -\frac{2}{3} x - 6 \)
, demonstrates that for every value of \(x\), there is not just one corresponding value of \(y\), but a multitude of values that satisfy the condition \((y\) is greater than or equal to \( -\frac{2}{3}x - 6 \)). It's like setting up a fence where all the points on one side are possible solutions. The inequality symbol '\(\geq\)' tells us that the solutions include both points on the boundary line \((y = -\frac{2}{3}x - 6)\) and all the points above it.

The substitution method is a straightforward technique used in algebra to determine if a particular point lies within the solution set of an inequality or equation. Following our exercise, let's break down the substitution step by step:

Step 1: Take the point you're investigating, in this case, \( (0,0) \), which represents an \(x-value\) of 0 and a \(y-value\) of 0.
Step 2: Substitute these values into the inequality in place of \(x\) and \(y\), yielding \( 0 \geq -\frac{2}{3}(0) - 6 \).
Step 3: Simplify the inequality by multiplying and adding the numbers, which here simplifies to \( 0 \geq -6 \).
By substituting, we've transformed the inequality into a statement that's easier to interpret and can quickly tell us whether our initial point is a solution.

When tackling an inequality, the goal is to figure out the 'solution set', which contains all the possible answers that satisfy the inequality's condition. In the context of our exercise, we've simplified the inequality to \( 0 \geq -6 \), which compares two numbers. Since 0 is indeed greater than -6, the inequality holds true, which means the point \( (0,0) \) is part of the solution set. It's important to note that inequalities can have many solutions, unlike equations which usually have a fixed set of solutions.