Problem 67 - College Algebra Essentials

Question

Write the given sentences as a system of inequalities in two variables. Then graph the system. The sum of the \(x\) -variable and the \(y\) -variable is no more than 2. The \(y\) -variable is no less than the difference between the square of the \(x\) -variable and 4

Step-by-Step Solution

Answer

The system of inequalities is given by \[ \begin{cases} x + y ≤ 2 \ y ≥ x² - 4 \end{cases}\] and its graphical representation is a plot where the area under the line \(x + y = 2\) and above the curve \(y = x² - 4\) are shaded. The intersection area represents the solution to the system.

Step 1: Translate Sentences into Inequalities

The two sentences can be written in mathematical language as follows: 'The sum of the \(x\) -variable and the \(y\) -variable is no more than 2' becomes \(x + y ≤ 2\). 'The \(y\) -variable is no less than the difference between the square of the \(x\) -variable and 4' becomes \(y ≥ x² - 4\). So, the system of inequalities is \[ \begin{cases} x + y ≤ 2 \ y ≥ x² - 4 \end{cases}\]

Step 2: Graph the Inequalities

To graph these inequalities, two different shaded zones need to be added to the Cartesian plane to represent where each condition is satisfied. For \(x + y ≤ 2\), a downward sloping line is drawn from the point (2,0) to (0,2) and everything beneath the line including the line itself gets shaded. For \(y ≥ x² - 4\), an upward-opening parabola is drawn with vertex at (0,-4) and everything above the parabola including the parabola itself gets shaded. The overlapping area represents the set of solutions for the system of inequalities.

Key Concepts

Algebraic Inequalities

An algebraic inequality, at its core, is a mathematical expression involving variables where one side is not necessarily equal to the other. Instead, a relation of being greater than, greater than or equal to, less than, or less than or equal to is used between two algebraic expressions. For example, the inequality (x + y ≤ 2) represents relation where the sum of the variables x and y is less than or equal to 2. To solve these inequalities, one must perform similar operations as with algebraic equations, while paying special attention to the direction of the inequality, especially when multiplying or dividing by negative numbers, as this will flip the inequality sign.

Understanding how to translate phrases into algebraic inequalities is crucial. Phrases like 'no more than' translate to 'less than or equal to' (≤), while 'no less than' would be 'greater than or equal to' (≥). Solidifying this mathematical language underpins successfully solving algebraic challenges and real-world problems alike.

Graphing Inequalities

The visual representation of inequalities on a graph allows us to see all possible solutions at once, which is essential when dealing with systems of inequalities. When graphing an inequality like (x + y ≤ 2), we start by graphing the associated equation (x + y = 2) as a boundary line. This line is drawn between the intercepts on the coordinate plane, and we use a shaded area to indicate where the inequality holds true. For this inequality, everything below and including the line is shaded to signify 'less than or equal to'.

With a quadratic inequality such as (y ≥ x² - 4), we graph the equation (y = x² - 4) as a parabola. When (y) is greater than or equal to something, the area above the graph of the equation is shaded. These visual aids help us to quickly see the intersection or overlap of shaded regions, which gives us the solution set where both inequalities are true at the same time.

Quadratic Functions

Quadratic functions are one of the most common and fundamental function types in algebra, taking the form (f(x) = ax² + bx + c), where a, b, and c are constants, and a ≠ 0. The graph of a quadratic function is a parabola that opens upwards if a > 0 and downwards if a < 0. The highest or lowest point, called the vertex, is a significant feature of the parabola and can be used to determine the function's minimum or maximum value.

The function in our inequality, (y = x² - 4), represents a basic quadratic with a = 1, b = 0, and c = -4, thus the parabola opens upwards and the vertex lies at (0, -4). In terms of inequalities, the quadratic function defines the boundary of the set of points that satisfy the inequality, and it is crucial to determine this boundary accurately when graphing the solution to a system involving a quadratic inequality.