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Chapter 2: Problem 88

Solve using the addition and multiplication principles. $$ 2>9-\frac{x}{5} $$

Short Answer

Expert verified
x > 35

Step by step solution

01

- Isolate the variable term

To isolate the term with the variable, subtract 9 from both sides of the inequality: \( 2 - 9 > 9 - \frac{x}{5} - 9 \) This simplifies to: \( -7 > - \frac{x}{5} \)
02

- Eliminate the fraction

Multiply both sides of the inequality by -5 to eliminate the fraction. Remember, multiplying or dividing by a negative number reverses the inequality sign: \( -7 \times (-5) < - \frac{x}{5} \times (-5) \) Simplifying this, we get: \( 35 < x \)
03

- Write the final solution

Rewrite the inequality to state the solution: \( x > 35 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

addition principle
The addition principle is a fundamental concept used in solving inequalities. It states that you can add or subtract the same value from both sides of an inequality without changing its meaning.
In our example, we need to isolate the term with the variable. The initial inequality is: \[ 2 > 9 - \frac{x}{5} \]
To isolate the variable term, we subtract 9 from both sides:
\[ 2 - 9 > 9 - \frac{x}{5} - 9 \]
This simplifies to:
\[ -7 > - \frac{x}{5} \]
By subtracting the same number from both sides, we maintain the inequality's balance.
multiplication principle
The multiplication principle involves multiplying or dividing both sides of an inequality by the same non-zero number. This helps in simplification, especially when dealing with fractions. However, remember a crucial rule: If you multiply or divide by a negative number, the direction of the inequality sign reverses.
In our example, we need to eliminate the fraction -\frac{x}{5}. We do this by multiplying both sides by -5. Recognize that this multiplication by a negative number will flip the inequality sign:
\[ -7 \times -5 < - \frac{x}{5} \times -5 \]
Simplifying, we get:
\[ 35 < x \]
Thus, we've effectively removed the fraction and solved for the variable.
inequality reversal
One critical aspect of solving inequalities is the reversal principle. When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.
In our example, we multiplied both sides by -5 to eliminate the fraction. Doing so we faced:
\[ -7 \times -5 < - \frac{x}{5} \times -5 \]
Remember, multiplying by -5 reverses the '>' sign to '<'. Thus, we have
\[ 35 < x \], which translates to
\[ x > 35 \]
Ensuring to reverse the inequality sign is vital whenever you deal with negative multiplications or divisions.
fraction elimination
Fractions can make inequalities look complicated. The goal is to eliminate the fraction to simplify the equation.
In our case, the inequality
\[ - \frac{x}{5} \]
contains a fraction. To eliminate it, we multiply every term by the denominator's value, which is -5. Doing this simplifies the inequality:
\[ - \frac{x}{5} \times -5 \]
equals
\[ -x \], resulting in the simplified inequality showing:
\[ 35 < x \]
or rewritten as:
\[ x > 35 \]. This step ensures that we get rid of fractions and make solving easier.

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Most popular questions from this chapter

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