91Ó°ÊÓ

When working with inequalities, the reciprocal of a fraction is a key tool to isolate the variable.
The reciprocal of a number is simply turning the fraction upside down.
For example, the reciprocal of \(\frac{16}{3}\) is \(\frac{3}{16}\). Using reciprocals helps us undo multiplication and clear fractions from our expressions.
This is crucial when solving inequalities as it helps us focus on the variable itself.
Remember that a reciprocal will always be such that the product of the original fraction and its reciprocal equals 1.In this inequality exercise, since \(\frac{16}{3}c\), the reciprocal \(\frac{3}{16}\) was used.
This is because applying it maintains balance while simplifying expressions, paving way to solve for \(c\).

Isolating the variable is all about getting the variable you want to solve for by itself on one side of an equation or inequality.
The main goal is to simplify the expression as much as possible around the variable.
This means you want the variable on one side and everything else on the other.In our problem, the variable is \(c\).
To isolate \(c\), it's necessary to get rid of the fraction and other coefficients affecting it. Steps to isolate the variable:

• Identify the variable you need to isolate.
• Use algebraic operations to move other terms to the other side.
• If the variable is being multiplied or divided by a fraction, use the reciprocal.

With \(\frac{16}{3}c \geq -48\), multiplying by \(\frac{3}{16}\) isolates \(c\) on its own.

Multiplying both sides of an inequality by the same number is an essential rule in algebra.
This technique maintains the equality or inequality, allowing us to simplify without changing the solution's meaning.
It ensures that whatever change applied to one side is equally applied to the other.In solving the inequality \(\frac{16c}{3} \geq -48\), both sides are multiplied by \(\frac{3}{16}\).
This action eliminates the fraction, simplifying the problem.Why multiply the whole inequality?

• To clear fractions or decimals and make calculations easier.
• To maintain and balance the equation or inequality.
• To help isolate the variable you're trying to solve for.

By consistently applying multiplication across both sides, you ensure the correctness of the solution process.

When it comes to inequalities, the goal is to find all possible values that satisfy the inequality condition.
The process is similar to solving equations but requires special attention when multiplying or dividing by negative numbers.In the example \(\frac{16c}{3} \geq -48\), after using the reciprocal and isolating \(c\), we determine the solution.
However, maintaining direction is crucial, and that means:

• If you multiply or divide both sides by a positive number, the inequality direction stays the same.
• On the other hand, using a negative number requires flipping the inequality sign.

Solving inequalities often involves:

• Step-by-step isolation of the variable.
• Careful management of inequalities so the solution remains valid.
• Checking the answer by substituting back into the original inequality.

In this problem, we found that \(c \geq -9\), which means for all values of \(c\) equal to or greater than -9, the inequality holds true.