Problem 45 - Algebra and Trigonometry

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} {y=x^{2}-1} \\ {x-y \geq-1} \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of inequalities is the region on or above the parabola \(y = x^2 - 1\) and also below the line \(y = x + 1\). These regions merge from the points (-1,0) and (1,0) and below in the demarcated x-axis area.

Step 1: Graph the Parabolic Equation

We can start by graphing the equation \(y = x^2 - 1\). This is a parabola with a vertex at (0,-1) and opens upwards. The x-intercepts of the parabola are at (-1, 0) and (1, 0). Refer to the graph.

Step 2: Graph the Linear Inequality

Next, we will graph the inequality \(x - y \geq -1\). We solve for y to get \(y \leq x + 1\). The boundary line \(y = x + 1\) is a straight line that passes through the points (0,1) and (-1,0). Which, coincidentally, are the x-intercepts of the parabola from Step 1. Because of the \(\leq\) symbol, the area of the graph below the straight line, including the line itself, will be shaded.

Step 3: Determine The Overlapping Region

The solution to the system of inequalities is the area where the shading from both the graphs matches. Observe the regions shaded by both the inequalities. The common region is the solution to the system.

Key Concepts

Graphing Inequalities

Graphing inequalities involves two main steps: graphing the corresponding equation as if it were an equality, and then identifying the region of the graph that satisfies the inequality. To start, always convert inequalities to equations by replacing symbols such as "<", "<=" with "=". This will give you the boundary line or curve.

• For linear inequalities, graph the equation as a straight line. You decide whether to make this line solid or dashed. Solid lines indicate "≤" or "≥" which include the line itself, while dashed lines show "<" or ">" which means the line is not included.
• For parabolic equations, graph the boundary as a curve. The parabola's shape gives the equation, while the inequality tells you which region to shade.

When shading, a quick way to check which side of the line or curve to shade is to pick a test point (usually (0,0) if it's not on the line) and see if it satisfies the inequality. If the point fits in the inequality, you shade the region containing that point.

Parabolic Equations

A parabolic equation generally takes the form of \(y = ax^2 + bx + c\). In this exercise, the parabola is \(y = x^2 - 1\). Here,

• "\(a\)" determines the direction of the parabola's opening: upwards if \(a > 0\) or downwards if \(a < 0\).
• The vertex is the turning point of the parabola. It's the lowest point if the parabola opens upwards or the highest if it opens downwards. In \(y = x^2 - 1\), the vertex is at (0,-1).
• The graph's intercepts can be found by setting \(x\) to zero to find \(y\)-intercepts, or \(y\) to zero to find \(x\)-intercepts.

The parabola divides the plane into two regions. The inequality \(y < x^2 - 1\), for example, would have us shade the region below the parabola, as it represents all points where \(y\) values are less than those on the parabola itself.

Linear Inequalities

Linear inequalities involve expressions of the form \(ax + by \,<, \,>, \,≤, or \,≥ \,c\). The key steps are to graph the boundary line and then figure out the correct region to shade.

• Reorder the inequality in the form \(y \leq mx + b\) or \(y \geq mx + b\) to quickly graph. The "m" represents the slope and "b" is the y-intercept.
• Draw the boundary line based on the converted equation. Again, remember solid lines include points on the boundary \(≤, ≥\) while dashed lines do not \(<, >\).
• Identify a test point that is not on the boundary. Plug it into the inequality. If true, shade the side containing the test point.

In the exercise, transforming \(x - y \geq -1\) to \(y \leq x + 1\) allowed the graphing of the line \(y = x + 1\). The shaded region includes all points below this line, including the line itself.