Problem 67 - Algebra 2

Question

Determine whether the given point satisfies each inequality. \(7 x+4 y \geq-15 ;(-4,2)\)

Step-by-Step Solution

Answer

The point (-4, 2) does not satisfy the inequality.

Step 1: Identify the inequality

We are given the inequality: \(7x + 4y \geq -15\). Our task is to determine if the point \((-4, 2)\) satisfies this inequality.

Step 2: Substitute the point into the inequality

Substitute \(x = -4\) and \(y = 2\) into the inequality. This gives us: \(7(-4) + 4(2) \geq -15\).

Step 3: Simplify the left side of the inequality

Calculate \(7(-4)\) which is \(-28\), and \(4(2)\) which is \(8\). So, we have: \(-28 + 8\).

Step 4: Evaluate the expression

Simplify \(-28 + 8\), which equals \(-20\). Now, our expression is \(-20 \geq -15\).

Step 5: Check if the inequality holds

Since \(-20\) is not greater than or equal to \(-15\), the inequality \(-20 \geq -15\) does not hold true.

Key Concepts

Coordinate Geometry

Coordinate Geometry is a method that allows us to understand geometric shapes in a numerical context. It uses the coordinate plane, which is defined by two number lines, commonly known as the x-axis and the y-axis, intersecting at the origin point \(0, 0\).
The conjunction of these axes helps us in locating and working with various points, lines, and shapes by assigning them numerical values called coordinates.

• The x-coordinate (first number) tells us how far to move horizontally from the origin.
• The y-coordinate (second number) tells us how far to move vertically from the origin.

For example, the point \((-4, 2)\) means moving 4 units to the left and 2 units upward from the origin. This systematic method serves as a foundation for analyzing inequalities involving linear equations in two variables.

Substitution Method

The Substitution Method is a technique used to solve equations by substituting given values into the equations. It makes it possible to check whether a point lies on a given line or curve defined by an equation.
In the context of inequalities, like \(7x + 4y \geq -15\), we substitute the given coordinates of the point into the inequality:

• Replace \(x\) with \(-4\)
• Replace \(y\) with \(2\)

This substitution transforms the inequality into a simple numerical expression that can be evaluated, helping us determine whether the specific point satisfies the original inequality.

Simplifying Expressions

Simplifying expressions is an essential step in solving mathematical problems. It involves breaking down complex expressions into their simplest form, making them easier to evaluate.
In our example, after substituting the values into the inequality \(7x + 4y \geq -15\), we obtain \(7(-4) + 4(2)\geq -15\).
From here, we perform each operation:

• Calculate \(7(-4)\) to get \(-28\).
• Calculate \(4(2)\) to get \(8\).

Then, we combine these results, giving us: \(-28 + 8\), which further simplifies to \(-20\).
Simplification helps us achieve a clearer numerical comparison, crucial for evaluating inequalities like ours.

Evaluating Inequalities

Evaluating inequalities involves determining whether a specific statement involving \(<, \leq, >, \geq\) is true or false.
Once the expression \(-20\geq -15\) is obtained, it is analyzed to check if the statement holds.
To evaluate

• Compare the numbers: Determine if \(-20\) is indeed greater than or equal to \(-15\).
• If true, the point satisfies the inequality. If false, it does not satisfy the inequality.

In our case, \(-20\) is not greater than nor equal to \(-15\), hence the point does not satisfy the original inequality. Understanding how to make these comparisons is fundamental in determining the relationships described by inequalities.