Problem 65 - Algebra 2

Question

Determine whether the given point satisfies each inequality. \(3 x+2 y \leq 10 ;(2,-1)\)

Step-by-Step Solution

Answer

The point (2, -1) satisfies the inequality 3x + 2y ≤ 10.

Step 1: Identify the Given Point

The point given is \((2, -1)\). Here, \(x = 2\) and \(y = -1\). We need to determine if this point satisfies the inequality \(3x + 2y \leq 10\).

Step 2: Substitute the Point into the Inequality

Substitute \(x = 2\) and \(y = -1\) into the inequality: \[3(2) + 2(-1) \leq 10\]

Step 3: Calculate the Left-hand Side of the Inequality

Calculate \(3 \times 2 + 2 \times (-1)\). First, multiply: \(3 \times 2 = 6\) and \(2 \times (-1) = -2\).

Step 4: Simplify the Expression

Add the results from Step 3. Thus, \(6 + (-2) = 4\). The inequality now is \(4 \leq 10\).

Step 5: Verify the Inequality

Check if \(4 \leq 10\) is true. Since 4 is indeed less than or equal to 10, the inequality is satisfied by the point \((2, -1)\).

Key Concepts

Substituting Values

Understanding how to substitute values is essential when working with inequalities. In this example, we need to substitute the values of the given point into the inequality. The point given is \[(2, -1)\], with \(x = 2\) and \(y = -1\). When substituting values, simply replace each variable in the inequality with its corresponding value. For this problem, we substitute into the inequality: \[3x + 2y \leq 10\]As follows:

• Replace \(x\) with 2: \(3(2)\)
• Replace \(y\) with -1: \(2(-1)\)

Thus, the inequality becomes: \[3(2) + 2(-1) \leq 10\]. This step ensures that we are working with specific numbers, making it easier to compute the expression and verify the inequality.

Verification of Inequalities

Verifying inequalities involves checking whether the calculated values satisfy the condition of the inequality. Once we've substituted the values into the inequality, the next step is to compute both sides of the inequality equation and ensure that the inequality holds true.From our substitution, we have: \[3(2) + 2(-1) \leq 10\].First, compute each multiplication:

• \(3 \times 2 = 6\)
• \(2 \times -1 = -2\)

Now, we add these results to simplify: \[6 + (-2) = 4\].The expression becomes \(4 \leq 10\). Checking this inequality is straightforward. Since 4 is indeed less than or equal to 10, we confirm that the point \((2, -1)\) satisfies the inequality. This verification acts as a proof of the point meeting the original condition set by the inequality.

Simplifying Expressions

Simplifying expressions involves reducing a complex expression into a simpler, more manageable form. This process is crucial in solving inequalities, as it allows us to understand and verify them more easily. After substituting the values, the expression within the inequality becomes \[3(2) + 2(-1)\]. To simplify the expression, follow these steps:

• Calculate \(3 \times 2\) resulting in 6.
• Calculate \(2 \times (-1)\) resulting in -2.

Next, combine the results of these operations: \[6 + (-2) = 4\]. This simplification helps us see that the left side of the inequality is 4, making it easier to compare with the right side, which is 10. By simplifying the expression, we ensure that we correctly evaluate whether the point satisfies the inequality \(4 \leq 10\). Effective simplification leads to a clearer understanding of not just whether inequalities hold true, but why they do so.