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Chapter 5: Problem 33

For the following problems, solve the inequalities. $$ \frac{10 x}{3} \leq 4 $$

Short Answer

Expert verified
Answer: The inequality holds true for any value of x such that \(x \leq \frac{6}{5}\).

Step by step solution

01

Multiply both sides of the inequality by the denominator

To eliminate the fraction, we need to multiply both sides of the inequality by the denominator, which is 3 in this case. $$ 3 \cdot \frac{10x}{3} \leq 3 \cdot 4 $$
02

Simplify both sides of the inequality

Now, we will simplify the left side of the inequality by cancellation property: $$ 10x \leq 12 $$
03

Isolate the variable x

To find the range of possible values for x, we need to isolate x by dividing both sides of the inequality by 10: $$ \frac{10x}{10} \leq \frac{12}{10} $$ Simplifying the inequality, we get: $$ x \leq \frac{6}{5} $$ So, the inequality is true for any value of x that is less than or equal to \(\frac{6}{5}\).

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Key Concepts

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

Fraction Simplification
Simplifying fractions is an essential skill in algebra. It involves reducing a fraction to its simplest form by eliminating common factors from the numerator and the denominator. In the given exercise, the fraction \( \frac{10x}{3} \) is simplified by removing the fraction altogether.

This is achieved by multiplying both sides of the inequality by 3, which is the denominator of the fraction. When we multiply \( \frac{10x}{3} \) by 3, the 3s in the numerator and denominator cancel each other out:
  • Start with \( 3 \times \frac{10x}{3} \).
  • The 3 in the numerator and the denominator "cancel," leaving you with \( 10x \).

This simplification is vital as it paves the way to isolate the variable and solve the inequality.
Multiplication Property of Inequalities
The multiplication property of inequalities allows you to multiply both sides of an inequality by the same positive number without changing the direction of the inequality. This property is crucial for removing fractions in inequalities by getting rid of the denominator.

For example, in the exercise, multiplying both sides of the inequality by 3 (a positive number) maintains the inequality's direction. This results in:
  • \( 3 \times \frac{10x}{3} \leq 3 \times 4 \)
  • Simplifies to \( 10x \leq 12 \)

Remember, when dealing with negative numbers, the direction of the inequality changes. However, in this context, since the multiplication was by a positive number, the inequality remains "less than or equal to."
Variable Isolation
Variable isolation involves manipulating an equation or inequality to solve for the variable of interest, often making it the subject. In this problem, we want to solve for \( x \), so we need to isolate it on one side of the inequality.

With \( 10x \leq 12 \), isolation requires dividing both sides by 10. When we do this division, we're effectively performing the opposite operation to undo the multiplication by 10:
  • \( \frac{10x}{10} \leq \frac{12}{10} \)
  • This simplifies to \( x \leq \frac{6}{5} \)

This means that \( x \) is less than or equal to \( \frac{6}{5} \), which gives the range of possible values that satisfy the inequality. Understanding variable isolation is key to mastering algebraic manipulations in solving equations and inequalities.

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Most popular questions from this chapter

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