Problem 66 - Algebra 2

Question

Determine whether \((0,0)\) satisfies each inequality. Write \(|x|+y \leq 3\)

Step-by-Step Solution

Answer

Yes, \((0,0)\) satisfies the inequality \(|x| + y \leq 3\).

Step 1: Identify the Inequality

In this step, we focus on understanding the given inequality, which is \(|x| + y \leq 3\). This inequality represents all points \((x,y)\) on the coordinate plane where the absolute value of \(x\) added to \(y\) results in a value that is less than or equal to 3.

Step 2: Substitute Values

Substitute the coordinates of the point \((0,0)\) into the inequality. In this case, \(x = 0\) and \(y = 0\). The inequality becomes \(|0| + 0 \leq 3\).

Step 3: Simplify the Expression

Evaluate the expression \(|0| + 0\). Since the absolute value of 0 is 0, the expression simplifies to \(0 + 0 = 0\).

Step 4: Compare with the Inequality

Compare the simplified result with the inequality. The inequality becomes \(0 \leq 3\), which is true, as 0 is indeed less than or equal to 3.

Key Concepts

Absolute Value

The absolute value of a number, denoted by two vertical bars like \(|x|\), is the distance of that number from zero on the number line. It is always a non-negative value. This concept is crucial in understanding and working with inequalities involving absolute values.
Absolute values are used to measure how far a number is from zero, regardless of direction. Think of it like measuring a distance that is always positive:

• For any positive number, say \(x = 5\), the absolute value is \(|5| = 5\).
• For a negative number, like \(x = -3\), the absolute value is \(|-3| = 3\).
• For zero, \(|0| = 0\), since it is neither positive nor negative.

In the exercise, using absolute value helps to handle any negative signs that may arise from the \(x\) coordinate. Only the magnitude or size of \(x\) affects the inequality, not its sign.

Coordinate Plane

The coordinate plane is a two-dimensional surface where each point is identified by a pair of numbers. These numbers, called coordinates, consist of:

• The horizontal value, the x-coordinate.
• The vertical value, the y-coordinate.

These coordinates take the form of \( (x, y) \) and allow for precise plotting of points. The origin, where \(x = 0\) and \(y = 0\), is denoted by \( (0, 0) \). This point is central to the plane and serves as a reference.
Understanding the coordinate plane is essential for solving inequalities as it helps visualize the solution set. For instance, when determining if \( (0, 0) \) satisfies an inequality, you place this point in the context of the plane and see if it falls inside the region described by the inequality. In this exercise, we assessed whether \( (0,0) \) satisfies the expression \(|x| + y \leq 3\) by substituting these coordinates and evaluating the result.

Evaluating Expressions

Evaluating expressions is the process of replacing variables with specific values and simplifying the result. In mathematical problems, this involves:

• Substituting given values for each variable.
• Carrying out arithmetic operations.
• Simplifying the expression to reach a final numerical result.

For the inequality \(|x| + y \leq 3\), we set \(x = 0\) and \(y = 0\) because we're testing the coordinates \( (0,0) \). This leads to the expression \(|0| + 0 = 0\). Evaluating, we find that \(0 \leq 3\), meaning the expression holds true.
By understanding how to evaluate expressions, one can check if certain points satisfy inequalities. It is a straightforward process of substitution and arithmetic, crucial in solving mathematical problems efficiently and accurately.