Problem 69 - College Algebra

Question

In Exercises 69–70, rewrite each inequality in the system without absolute value bars. Then graph the rewritten system in rectangular coordinates. $$\left\\{\begin{array}{l} |x| \leq 2 \\ |y| \leq 3 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of inequalities is the area where -2 ≤ x ≤ 2 and -3 ≤ y ≤ 3 intersect in the rectangular coordinate system.

Step 1: Interpreting absolute value inequalities

The absolute value |x| is the distance of x from the origin on the number line. Therefore, |x| ≤ 2 means that x is a distance of 2 or less from the origin. This translates into the standard inequality as -2 ≤ x ≤ 2. \nSimilarly, |y| ≤ 3 means that y is a distance of 3 or less from the origin. This translates into the standard inequality as -3 ≤ y ≤ 3.

Step 2: Graphing the inequalities

Next, these inequalities will be graphed on the rectangular coordinate system. First, draw a number line for x from -2 to 2 and another number line for y from -3 to 3. These will create a rectangle. \nThe solution to -2 ≤ x ≤ 2 will be a vertical strip (a line at x=-2 to a line at x=2). \nSimilarly, the solution to -3 ≤ y ≤ 3 will be a horizontal strip (a line at y=-3 to a line at y=3). \nThe area where both conditions are satisfied (both vertical and horizontal strip overlap) is the solution to the system of inequalities and should be shaded.

Key Concepts

Graphing Inequalities

When working with inequalities, graphing provides a visual representation of the solution set. **Graphing inequalities** involves shading the region that satisfies the inequality on the coordinate plane. Consider inequalities like \(-2 \leq x \leq 2\) and \(-3 \leq y \leq 3\). This means that the 'x' values are restricted to the range between -2 and 2, while 'y' values fall between -3 and 3.
The region that satisfies both inequalities will be where these ranges overlap on the plane. For the mentioned inequalities, the resulting graph would form a rectangle. The shaded area within the rectangle represents all the coordinates (x, y) that fulfill the conditions. This visual can reinforce the understanding of how inequalities operate and their solutions.
A quick way to check your work is by testing points. If a point lies inside the shaded area, each separate inequality must be true. This helps validate that your graph accurately represents the solution.

Rectangular Coordinate System

The **rectangular coordinate system** is a two-dimensional plane characterized by the 'x' axis (horizontal) and the 'y' axis (vertical). This system is often referred to as the Cartesian coordinate system. It is foundational for graphing as it allows you to plot points, lines, and shapes corresponding to algebraic equations and inequalities.
Each point on the plane is determined by an ordered pair (x, y), where 'x' represents the horizontal position and 'y' the vertical position. For instance, the point (2, -3) indicates a position 2 units to the right on the x-axis and 3 units down on the y-axis from the origin (0,0).
Within this system, inequalities are not just simple lines or points but areas. The coordinate system helps us visualize where these areas lie, thereby solving and understanding inequality systems better. It transforms abstract algebraic expressions into tangible images.

Solving Inequality Systems

Solving systems of inequalities involves finding a set of values that satisfies all the given conditions at once. For instance, if presented with \(|x| \leq 2\) and \(|y| \leq 3\), you'll first rewrite these to standard inequalities like \(-2 \leq x \leq 2\) and \(-3 \leq y \leq 3\).
Each of these inequalities can be thought of as strips on their respective axes: a vertical strip for the 'x' inequality and a horizontal strip for the 'y' inequality. Solving the system involves identifying the intersection of these strips.
The solution, in this case, would be the rectangle in the middle of the coordinate plane formed by these intersecting strips. The logic is that any point within this rectangle satisfies both the x and y constraints simultaneously, thus solving the inequality system. When visualized, it becomes much clearer how these constraints interact and overlap.