Question

MAKING AN ARGUMENT You claim the system of inequalities below, where \(a\) and \(b\) are real numbers, has no solution. Your friend claims the system will always have at least one solution. Who is correct? Explain. $$ \begin{aligned} &y<(x+a)^2 \\ &y<(x+b)^2 \end{aligned} $$

Short answer

Your friend's claim is correct, a solution to the system of inequalities will always exist, regardless of the real numbers 'a' and 'b'. The point lying at the bottom of the lowest parabola satisfies both conditions.

Step-by-step solution

Understanding the inequalities

The problem presents two parabolas with the forms \(y < (x+a)^2\) and \(y < (x+b)^2\). The first parabola is shifted 'a' units to the left or right depending on whether 'a' is positive or negative. The second is shifted 'b' units likewise. For a point \((x, y)\) to be a solution, it needs to lie below both parabolas, meaning the 'y' value must be less than the 'y' value of both parabolas at that 'x' value.

Debating over the solutions

The claim is that no solution exists for these inequalities. However, no matter the real numbers 'a' and 'b' chosen, it can be seen that the point at the bottom of the lowest parabola (say, without loss of generality, at \((b, b^2)\)) is a solution to both inequalities for any 'x' value, regardless of 'a' and 'b'.

Providing an example

To prove this argument correct, a pair of 'a' and 'b' can be taken. Consider, for example, 'a' = -2 and 'b' = 2. Then the system of inequalities becomes \(y < (x-2)^2\) and \(y < (x+2)^2\), implying the point (2, 4) will be a solution for both inequalities, hence a solution for the system always exists.

Key concepts

Algebraic Reasoning

Algebraic reasoning is the process of solving problems by manipulating algebraic expressions and equations through the application of mathematical principles. It is an essential skill when working with a system of inequalities, which consists of multiple inequalities that must be satisfied simultaneously.

In the context of the provided exercise, we're tasked with evaluating the assertion regarding the existence of solutions to a given system of inequalities. Using algebraic reasoning, we understand that the solution to such a system must satisfy all the inequalities at the same time. It requires us to consider the relations between the algebraic expressions involved, in this case, two quadratic equations defining parabolas, and to identify whether there is a common area that satisfies both inequalities.

Visualizing Relationships

When working through the exercise, one can visualize or plot the parabolas on a graph to better understand the relationship between them. By doing so, it's evident that the distinct parabolas will intersect or be parallel to each other, this visual aid aids the algebraic reasoning process. Through systematic exploration of the intersections and relative positions of the parabolas, students can derive the reasoning that leads us to conclude if a shared solution region exists.

Quadratic Inequalities

Quadratic inequalities, such as those given in the textbook exercise, are expressions where a quadratic function is set to be greater than or less than a certain value. In the standard form, a quadratic inequality looks like this: \[ ax^2 + bx + c > 0 \text{ or } ax^2 + bx + c < 0 \]where 'a', 'b', and 'c' are constants.Quadratic inequalities can tell us about the regions where the parabola associated with the quadratic function lies above or below a particular value on the y-axis. Solutions to these inequalities are the 'x' values that make the inequality true, and are often represented on a number line or on a Cartesian plane.

Examining the Intersection Area

In the given exercise, we examine where the parabolas denoted by \( y < (x+a)^2 \) and \( y < (x+b)^2 \) both lie below the 'y' value. This area of intersection is crucial and represents the set of all solutions to the system of inequalities. Algebraic reasoning aids in understanding that if any region of overlap exists between the two sets of solutions for the separate inequalities, then that region represents the set of solutions for the system of inequalities.

Parabolas

Parabolas are U-shaped graphs represented by a quadratic function in the form \( y = ax^2 + bx + c \). They are symmetric about a vertical line called the axis of symmetry and have a peak point called the vertex. In the given inequalities, the parabolas are defined as \( y < (x+a)^2 \) and \( y < (x+b)^2 \), indicating that we are interested in the region below these curves on the Cartesian plane.

Understanding the shape and position of parabolas is critical when solving systems of inequalities that involve them. The direction a parabola opens (upward or downward) depends on the sign of 'a'. If 'a' is positive, the parabola opens upward; if negative, it opens downward. In this problem, the parabolas both open upward as the leading coefficient is positive.

Shifting Parabolas

The 'a' and 'b' values in \( (x+a)^2 \) and \( (x+b)^2 \) represent horizontal shifts from the origin point of the standard parabola \( y = x^2 \). When 'a' and 'b' are real numbers, these shifts place the vertices of the parabolas at different positions along the x-axis. Identifying the vertex and axis of symmetry of each parabola offers significant insight when determining the set of points that satisfy both inequalities. Notably, the bottommost point of the lower parabola always provides at least one solution, as mentioned in the step-by-step solution.