Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

Given \(-7<-3\), determine the inequality obtained if (a) 5 is added to both sides (b) 4 is subtracted from both sides (c) both sides are multiplied by \(\frac{1}{3}\) (d) both sides are multiplied by \(-\frac{1}{3}\)

Step-by-Step Solution

Answer

Resulting inequalities: (a) \(-2 < 2\), (b) \(-11 < -7\), (c) \(-\frac{7}{3} < -1\), (d) \(\frac{7}{3} > 1\).

Step 1: Understanding the Original Inequality

The given inequality is \(-7 < -3\). This inequality simply tells us that negative seven is less than negative three.

Step 1: Adding 5 to Both Sides

When 5 is added to both sides of the inequality \(-7 < -3\), we get:\[-7 + 5 < -3 + 5\]This simplifies to:\[-2 < 2\]Thus, the inequality remains \(-2 < 2\).

Step 2: Subtracting 4 from Both Sides

Subtracting 4 from both sides of the inequality \(-7 < -3\) gives:\[-7 - 4 < -3 - 4\]Simplifying this, we have:\[-11 < -7\]The resulting inequality is \(-11 < -7\).

Step 3: Multiplying Both Sides by \(\frac{1}{3}\)

Multiplying both sides of \(-7 < -3\) by \(\frac{1}{3}\) gives:\[-\frac{7}{3} < -\frac{3}{3}\]Simplifying this results in:\[-\frac{7}{3} < -1\]Hence, the inequality becomes \(-\frac{7}{3} < -1\).

Step 4: Multiplying Both Sides by \(-\frac{1}{3}\)

The inequality \(-7 < -3\) is multiplied by \(-\frac{1}{3}\), requiring us to flip the inequality sign. This results in:\[-7(-\frac{1}{3}) > -3(-\frac{1}{3})\]Which simplifies to:\[\frac{7}{3} > 1\]The new inequality is \(\frac{7}{3} > 1\).

Key Concepts

Addition and Subtraction in Inequalities

Understanding how addition and subtraction affect inequalities is crucial. They are perhaps the simplest operations that you can perform on inequalities, and they **do not** change the direction of the inequality sign.
For instance, consider the inequality \(-7 < -3\).
Let's see what happens when you add or subtract a number from both sides of the inequality:

• Addition: Adding 5 to both sides results in \(-7 + 5 < -3 + 5\). This simplifies to \(-2 < 2\). Despite the arithmetic change, the inequality's direction remains unchanged.
• Subtraction: Subtracting 4 from both sides gives \(-7 - 4 < -3 - 4\). Simplifying, we have \(-11 < -7\). Again, the direction of the inequality is preserved.

In both scenarios, you can see that the inequality sign—that is, the relation between the two numbers—remains the same. This consistency is a defining characteristic of how addition and subtraction influence inequalities.

Multiplication of Inequalities

When dealing with the multiplication of inequalities, careful attention must be given to the sign of the number you are multiplying by.
If you multiply both sides of an inequality by a positive number, the inequality holds its direction.
This is what we observe in our given inequality \(-7 < -3\) when multiplied by \(\frac{1}{3}\):

• The computation steps include: \(-7 \cdot \frac{1}{3} < -3 \cdot \frac{1}{3}\). This simplifies to \(-\frac{7}{3} < -1\).

The inequality sign remains the same because \(\frac{1}{3}\) is a positive number.
So, when you multiply all parts of an inequality by a positive value, you keep the relational order of the inequality consistent.

Transformation of Inequalities

In transforming inequalities, we often apply arithmetic operations or algebraic manipulations to simplify or improve readability.
Transformation involves defining equivalent inequalities obtained through legitimate operations.
This is important to preserve the meaning and the truth of the original inequality.
For example, by multiplying both sides of the inequality \(-7 < -3\) by \(\frac{1}{3}\), we transform the inequality without altering its truth.

• We calculated: \(-\frac{7}{3} < -1\).

This transformation is as valid as the original statement and helps in expressing the inequality in simpler or more useful forms for problem solving.

Inequality Sign Reversal

The rule of inequality sign reversal is crucial when multiplying or dividing by a negative number.
This is because multiplying or dividing by a negative "flips" the entire numerical relationship.
In our example, multiplying \(-7 < -3\) by \(-\frac{1}{3}\) requires reversing the inequality sign:

• The process looks like this: \(-7(-\frac{1}{3}) > -3(-\frac{1}{3})\).
• This simplifies to \(\frac{7}{3} > 1\).

This flip is essential to understand, as ignoring it will lead to incorrect solutions.
Remembering to reverse the inequality sign whenever you multiply or divide both sides of an inequality by a negative number is vital for maintaining mathematical correctness.