Problem 22 Systems of Equations and Inequal... [FREE SOLUTION]

Chapter 8: Problem 22

Systems of Equations and Inequalities. $$y \geq x^{2}-1$$

Short Answer

Expert verified

The solution to the inequality $y \geq x^{2}-1$ is all the points (x, y) that lie above and on the curve of the parabola $y = x^{2}-1$.

Step by step solution

Understand the inequality

The inequality $y \geq x^{2} - 1$ is expressed in terms of $y$ and $x$. $x^{2} - 1$ represents a parabola, which opens upwards. Here, the value of 'y' is greater than or equal to 'x' squared minus one.

Graph the inequality

To graph $y \geq x^{2}-1$, first, plot the parabola of $y = x^{2}-1$. The vertex of this parabola is at the origin, and the parabola will open upwards. As the inequality is $y \geq$, the area above the curve $y = x^{2}-1$, including the curve, represents the solution to the inequality.

Define the solution

The solution of the inequality $y \geq x^{2}-1$ is the region that lies above the parabola $y = x^{2}-1$, including the curve. In other words, each point in that region makes the inequality true. Therefore, any point (x, y) that lies in this region is a solution to the inequality.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Inequalities

Graphing inequalities involves plotting a region on a coordinate plane that satisfies the inequality. In our exercise, the inequality is $ y \geq x^2 - 1 $. This inequality indicates that we need to graph all points where the value of $ y $ is either equal to or greater than $ x^2 - 1 $.

**Step 1: Plot the boundary**. First, we identify the curve $ y = x^2 - 1 $. This is the boundary line of our inequality, which forms a parabola. It's essential to draw this line precisely since it shows where $ y $ equals exactly $ x^2 - 1 $.
**Step 2: Determine the shading**. Since $ y \geq x^2 - 1 $, the area above the parabola is shaded. This shading includes the region and the boundary line itself because the inequality includes $ \geq $ (greater than or equal to).
**Step 3: Use test points**. If you're unsure about the shading direction, pick a test point not on the boundary line (like the origin, $ (0,0) $), and substitute it into the inequality to see if it holds true. This is a useful technique to confirm your shaded area.

Graphing inequalities visually represents solutions on a graph, making it easier to see all points that satisfy the inequality. It is crucial to understand and accurately represent both the boundary line and the correct region to depict the solution space.

Parabolas

A parabola is a U-shaped curve that can open either upwards or downwards. It is defined by a quadratic equation, such as $ y = x^2 - 1 $ in our example. Understanding the properties of parabolas is key to graphing them accurately.

**Vertex**: The vertex is the tip of the parabola. For $ y = x^2 - 1 $, the vertex is at $ (0, -1) $. It is a crucial point as it acts as the parabola's lowest point (for upward opening) or highest point (for downward opening).
**Direction**: The expression $ y = x^2 - 1 $ indicates an upward opening parabola because the $ x^2 $ term is positive. If the term was negative, the parabola would open downwards.
**Axis of Symmetry**: Parabolas have an axis of symmetry, a vertical line that passes through the vertex. For our parabola, this line is $ x = 0 $, which runs vertically through the vertex.

Parabolas are quadratic functions that can visually demonstrate solutions of equations and inequalities. They have distinct, predictable shapes that make them straightforward to plot once understood.

Solution of Inequalities

When solving inequalities, the objective is to identify all possible values that will make the inequality true. With $ y \geq x^2 - 1 $, we're interested in finding all pairs $ (x, y) $ that satisfy this condition.

**Identify the boundary**: First, determine the equality $ y = x^2 - 1 $. This equation helps to visualize the limit of the solutions. The boundary itself is included in the solutions because of the "equal to" part of $ \geq $.
**Understanding the region**: The inequality encompasses all **above** or **on** the parabola $ y = x^2 - 1 $. Any point in this region will satisfy the inequality.
**Multiple solutions**: Unlike equations with one or two solutions (like linear or quadratic equations), inequalities often have infinite solutions. These solutions form a region instead of discrete points.

Finding the solution to an inequality means finding a whole host of points that lie in a specified area, or along a curve. Proper graphing helps to visualize and understand these solutions better, demonstrating the region that fulfills the inequality conditions.

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91影视

Systems of Equations and Inequalities. $$y \geq x^{2}-1$$

Short Answer

Step by step solution

Understand the inequality

Graph the inequality

Define the solution

Key Concepts

Graphing Inequalities

Parabolas

Solution of Inequalities

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