What is a Solution Set in Math? Understanding Inequalities and Equations

In mathematics, a solution set is a crucial concept that refers to the collection of all values that satisfy a particular equation or inequality. To better understand this concept, let’s break it down using a specific example involving both an inequality and an equation.

Consider the two conditions: (1) $2x > -8$ and (2) $-5x + 7 = 12$. We will solve both to find the values of $x$ that meet both criteria.

**Step 1: Solving the Inequality**
Starting with the inequality $2x > -8$, we want to isolate $x$. To do this, divide both sides of the inequality by 2:

$$x > -4$$
This tells us that any value greater than $-4$ will satisfy the inequality.

**Step 2: Solving the Equation**
Next, we tackle the equation $-5x + 7 = 12$. First, subtract 7 from both sides:

$$-5x = 5$$
Now, divide both sides by -5:

$$x = -1$$
This means that $x$ must be equal to $-1$ for the equation to hold true.

**Combining Results**
Now, we need to find a value that satisfies both conditions simultaneously. From the inequality, we have $x > -4$, and from the equation, we know that $x = -1$.
Since $-1$ is indeed greater than $-4$, we find that it satisfies both the inequality and the equation.

**Conclusion**
Thus, the solution set for these conditions is simply $ ext{-1}$. This means that $-1$ is the only number that makes both the inequality and the equation true.

Understanding solution sets is essential not only in solving mathematical problems but also in real-world applications like optimization, where you might need to find feasible solutions that meet multiple constraints. Whether in economics, engineering, or everyday decision-making, knowing how to find solution sets can help you achieve your goals effectively.

If you have more questions or need further clarification, feel free to check out additional resources or ask for help in understanding different types of equations and inequalities!