91Ó°ÊÓ

An inequality, like the one we have here, is similar to an equation but instead of using equality ("="), it expresses a relationship where one side may be smaller, larger, or equal under particular circumstances. The original inequality given is \[-x - 3 \leq 4\].

The "\(\leq\)" sign suggests that whatever solution we find must make the left-hand side less than or equal to the right-hand side. Solving inequalities involves similar steps to solving equations. The primary goal is to isolate the variable, usually represented by \(x\), on one side.

• The operations performed on inequalities mirror those in equations - we can add, subtract, multiply, or divide.
• However, one critical difference is: when we multiply or divide both sides by a negative number, the inequality sign must be flipped.

Understanding this fundamental concept allows us to approach and solve inequalities systematically.

Breaking down the inequality into manageable steps simplifies the process:

Step 1: Remove Constant Term
We begin by eliminating the constant term from the side with the variable. In our example, we add 3 to both sides of the inequality:\[-x - 3 + 3 \leq 4 + 3\]. This results in \(-x \leq 7\).

Step 2: Solve for x
The next step is to isolate the variable, \(x\). Here, we need to divide by \(-1\) to solve for \(x\). Remember, dividing by a negative number reverses the inequality sign:\[-x \leq 7\] becomes \(x \geq -7\).

When you reverse the inequality sign, it reflects the change in relationship caused by multiplication or division through a negative number. This step-by-step solving approach ensures clarity and correctness in finding solutions to inequalities.

Verification adds confidence to your solution by testing its validity. After obtaining \(x \geq -7\) as our solution, we should check if it is indeed correct. Here's how:

Pick a value that fits within the solution range. For instance, let's choose \(x = -7\), a boundary value.

• Substitute \(-7\) back into the original inequality \(-(-7) - 3 \leq 4\).
• Solve the simplified expression: \[7 - 3 \leq 4\].
• This statement is true because \(4 \leq 4\).

By verifying with at least one point from the solution set, we confirm the correctness of our solution \(x \geq -7\). Always verify to ensure that no step in the inequality solving process has been overlooked or misapplied.