91Ó°ÊÓ

When working with inequalities, understanding how to handle the inequality sign is crucial.
Specifically, whenever you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign must be reversed.
In this exercise, after isolating the variable -z, we ended up with -z > -9.
To solve for z, we multiplied both sides by -1, which changed the inequality to z < 9.
This flip in the inequality direction is critical for maintaining the correctness of the solution.
It's essential to remember this rule, as overlooking it can lead to incorrect answers.
Think of it like flipping the sign when changing direction on a number line.
If you're not sure, you can always plug values back into the original inequality to check your work.
For instance, if z = 8, it would satisfy the inequality -z + 14 > 5, because -8 + 14 is greater than 5.

Isolating the variable is one of the first steps in solving any equation or inequality.
In this exercise, we start with -z + 14 > 5.
To isolate -z, we need to get rid of the +14 on the left side.
We do this by subtracting 14 from both sides: -z + 14 - 14 > 5 - 14, which simplifies to -z > -9.
The goal here is to have z, or the variable, on one side of the inequality.
This makes it easier to solve and understand.
Always ensure to perform the same operation on both sides of the inequality to maintain balance.
This approach is not just specific to inequalities but is also a key step in solving equations.

Multiplying by negative numbers can be tricky when dealing with inequalities.
The main rule to remember is that the inequality sign changes direction whenever you multiply or divide both sides of the inequality by a negative number.
In our exercise, after isolating the variable, we had -z > -9.
To solve for z, we multiply both sides by -1: -1(-z) < -1(-9).
Notice how the inequality sign reversed from > to <.
This step yields z < 9, which is our simplified answer.
Skipping this step or forgetting to reverse the sign can lead to incorrect solutions.
This rule stems from the properties of the number line and helps keep the inequality consistent with mathematical logic.
Always double-check your work to ensure you haven't missed this critical step.
Remember to apply this rule consistently in all your inequality problems.