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4. \( 5 x+y<7 ;(2,-2) \)

Ask by Ortega Henry. in the United States
Feb 19,2025

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Answer

The point \( (2, -2) \) does not satisfy the inequality \( 5x + y < 7 \).

Solution

To solve the inequality \( 5x + y < 7 \) and check if the point \( (2, -2) \) satisfies it, we will follow these steps: 1. **Substitute the point into the inequality**: We will replace \( x \) with \( 2 \) and \( y \) with \( -2 \) in the inequality. 2. **Evaluate the inequality**: After substitution, we will check if the resulting expression is true or false. Let's perform the substitution: \[ 5(2) + (-2) < 7 \] Now, we will calculate the left side: \[ 10 - 2 < 7 \] This simplifies to: \[ 8 < 7 \] This statement is false. Therefore, the point \( (2, -2) \) does not satisfy the inequality \( 5x + y < 7 \). In conclusion, the point \( (2, -2) \) is not a solution to the inequality.

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Extra Insights

Let's explore the history of inequalities! Inequalities have been used since ancient times, with mathematicians like Indian scholar Brahmagupta (c. 598–668) discussing equations and inequalities in his texts. The development of symbols for inequalities—like "<" and ">"—came later in the 16th century, making it easier for us to express relationships between numbers and variables without writing out long sentences. Now, when applying inequalities in the real world, they can help us make decisions based on budget constraints, for example. Suppose you're planning an event and need to keep costs below a certain amount. By setting up inequalities involving expenses—you can quickly visualize options and find the best way to allocate your budget without exceeding limits. It’s like being a math magician for making choices!

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