Problem 12 - Algebra Form and Function

Question

Each of the inequalities can be solved by performing a single operation on both sides. State the operation, indicating whether or not the inequality changes direction. Solve the inequality. $$ 27>-m $$

Step-by-Step Solution

Answer

Question: Solve the inequality and find the possible values of \(m\): \(-m > 27\). Answer: The solution to the inequality is \(m > -27\).

Step 1: Identify the operation needed to isolate the variable

In this case, the variable "m" is being multiplied by -1. To isolate "m", we need to divide both sides of the inequality by -1.

Step 2: Perform the operation and check if the inequality changes direction

When we divide (or multiply) both sides of an inequality by a negative number, the direction of the inequality changes. Since we are dividing by -1, the inequality will change direction: $$ \frac{27}{-1} < \frac{-m}{-1} $$

Step 3: Simplify the inequality

Now we simplify the inequality: $$ -27 < m $$

Step 4: Final Answer

The solution to the inequality is \(m > -27\).

Key Concepts

inequality direction change

When dealing with inequalities, you need to be careful because sometimes the direction of the inequality symbol needs to change.
An important rule to remember is: if you multiply or divide both sides of an inequality by a negative number, the direction of the inequality must change.
For example, if you start with the inequality \( a > b \) and you multiply both sides by \(-1\), it will change to \(-a < -b\). This is because reversing the sign of a number reflects it over zero on the number line, which changes the order.
Therefore, always ensure to flip the inequality symbol when multiplying or dividing by negative numbers so that the inequality stays true.

multiplication and division in inequalities

When solving inequalities, multiplication and division are common operations used to isolate variables.
Let's explore what happens when you use these operations with inequalities:

• Multiplying or dividing both sides by the **same positive number**: The direction of the inequality stays the same. For example, if \( c < d \), then \( c \times k < d \times k \) if \( k > 0 \).
• Multiplying or dividing both sides by the **same negative number**: The direction of the inequality changes. For instance, using \( 27 > -m \) and dividing by \(-1\) turns it into \( -27 < m \).
  This change occurs because negative multiplication reflects the ordering of numbers.

So, a good rule to remember is to flip the inequality sign when multiplying or dividing by a negative number. Otherwise, for positive numbers, the direction stays the same.

negative coefficients in inequalities

Negative coefficients can appear tricky when solving inequalities, but with careful attention, they become straightforward.
Consider situations where your variable has a negative coefficient, such as \( -m \) in \( 27 > -m \). Here, the goal is to isolate \( m \) by manipulating the inequality.
Usually, this involves dividing or multiplying by the negative coefficient:

• In our example, \( -m \) must be divided by \(-1\) to turn it into \( m \). This means both sides of the inequality \( 27 > -m \) are divided by \(-1\).

This operation results in changing the inequality's direction to maintain correctness. That's why \( 27 > -m \) becomes \( -27 < m \), or equivalently, \( m > -27 \).
Thus, when faced with negative coefficients, always ensure the solution steps preserve the inequality integrity by switching the inequality direction and isolating the variable correctly.