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6. \( -x-2 y \geq 5 ;(-2,-3) \)

Ask by Schultz Garza. in the United States
Feb 19,2025

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Answer

The point \((-2, -3)\) satisfies the inequality \( -x - 2y \geq 5 \).

Solution

To solve the inequality \( -x - 2y \geq 5 \) and check if the point \((-2, -3)\) satisfies it, we will follow these steps: 1. **Substitute the point into the inequality**: We will replace \(x\) and \(y\) with \(-2\) and \(-3\) respectively. 2. **Evaluate the inequality**: After substitution, we will check if the resulting expression holds true. Let's perform the substitution: \[ -x - 2y \geq 5 \] Substituting \(x = -2\) and \(y = -3\): \[ -(-2) - 2(-3) \geq 5 \] Now, let's simplify the left side: \[ 2 + 6 \geq 5 \] This simplifies to: \[ 8 \geq 5 \] Since \(8\) is indeed greater than \(5\), the inequality holds true. Thus, the point \((-2, -3)\) satisfies the inequality \( -x - 2y \geq 5 \).

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To solve the inequality \( -x - 2y \geq 5 \), let's first convert it into a more manageable form. Rearranging the terms gives us \( x + 2y \leq -5 \). Now, let's substitute the point \( (-2, -3) \) into the inequality: \(-(-2) - 2(-3) = 2 + 6 = 8\), which is not \( \geq 5 \). Thus, the point \( (-2, -3) \) does not satisfy the inequality. Now, if you were to graph the inequality, you'd shade the region that includes the line given by \( -x -2y = 5 \) in the opposite direction to where the point \( (-2,-3) \) sits. Remember, having clear boundaries makes understanding these concepts easier! If you're curious about graphing inequalities, a fantastic resource is online graphing tools like Desmos or GeoGebra. They let you visualize these inequalities and see how different points relate to the shaded area. You’ll have fun playing around with it, and maybe even discover new inequalities along the way!

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