Insert either , or = in the shaded area to make a true statement. 3040−34 □ 1415⋅1514\frac{30}{40} - \frac{3}{4} \ \square \ \frac{14}{15} \cdot \frac{15}{14} 4030−43 □ 1514⋅1415

Ch. P - Fundamental Concepts of Algebra

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Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

Problem 107

Chapter 1, Problem 107

Insert either <, >, or = in the shaded area to make a true statement. $\frac{30}{40}\) - $\frac{3}{4}$ \ $\square$ \ $\frac{14}{15}$ $\cdot$ \(\frac{15}{14}$

Verified step by step guidance

First, simplify the left side of the inequality: calculate $\frac{30}{40} - \frac{3}{4}$. Start by simplifying $\frac{30}{40}$ to its lowest terms.

Next, convert both fractions on the left side to have a common denominator so you can subtract them easily.

Then, simplify the right side of the inequality: calculate $\frac{14}{15} \cdot \frac{15}{14}$. Notice that multiplication of these fractions might simplify directly.

After simplifying both sides, compare the resulting values to determine whether the left side is less than, greater than, or equal to the right side.

Finally, insert the correct inequality symbol ($<$, $>$, or $=$) in the shaded area based on your comparison.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Comparing Fractions

To compare fractions, they must be expressed with a common denominator or converted to decimal form. This allows for a direct comparison of their sizes to determine which is greater, smaller, or if they are equal.

Order of Operations

The order of operations dictates the sequence in which mathematical operations are performed: parentheses, exponents, multiplication and division (left to right), then addition and subtraction (left to right). Correct application ensures accurate evaluation of expressions.

Simplifying Fractions and Products

Simplifying fractions involves reducing them to their lowest terms by dividing numerator and denominator by their greatest common divisor. Multiplying fractions requires multiplying numerators and denominators directly, followed by simplification to compare or evaluate expressions.

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Insert either <, >, or = in the shaded area to make a true statement. 3040−34 □ 1415⋅1514\(\frac{30}{40}\) - \(\frac{3}{4}\) \ \(\square\) \ \(\frac{14}{15}\) \(\cdot\) \(\frac{15}{14}\)4030−43 □ 1514⋅1415

Verified video answer for a similar problem:

Key Concepts

Comparing Fractions

Order of Operations

Simplifying Fractions and Products

Insert either <, >, or = in the shaded area to make a true statement. $\frac{30}{40}\) - \(\frac{3}{4}\) \ \(\square\) \ \(\frac{14}{15}\) \(\cdot\) \(\frac{15}{14}$