Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 7: Problem 67
Simplify. Graph \(y \leq \sqrt{x+1}\)
Short Answer
Expert verified
Graph the parabola and shade below it for \(x \geq -1\).
Step by step solution
01
Understanding the Inequality
The given inequality is an equation of the form \(y \leq \sqrt{x+1}\). This means that we are interested in the region where the \(y\)-value is less than or equal to the square root of \(x + 1\).
02
Consider the Equality for Boundary
To understand the boundary of the region, consider the equation \(y = \sqrt{x+1}\). This represents the boundary line of our inequality, which is a parabola opened sideways to the right.
03
Determine Domain and Range for the Equality
For \(y = \sqrt{x+1}\), \(x+1\) must be non-negative. Therefore, \(x \geq -1\). Since all square roots give non-negative results, \(y \geq 0\).
04
Graph the Boundary
Graph the equation \(y = \sqrt{x+1}\) for \(x \geq -1\). To do this, choose points such as \(x = -1\), \(x = 0\), and \(x = 3\), and calculate corresponding \(y\)-values: \((x, y) = (-1, 0), (0, 1), (3, 2)\). Connect these points and plot the curve.
05
Shade the Region
The inequality \(y \leq \sqrt{x+1}\) requires shading below the parabola. Test a point below the parabola, such as \((x, y) = (0, 0)\), which satisfies the inequality \(0 \leq \sqrt{0+1} = 1\), indicating this side needs shading.
06
Verify with Boundary Curve
Recheck that points on the parabola, like \((0, 1)\), satisfy the equality \(1 = \sqrt{0+1}\), confirming they are on the border between shaded and non-shaded areas.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Inequalities
Inequalities are mathematical expressions that describe a range of values that satisfy a particular condition. They differ from equations in that they show a relationship of being greater than, less than, or equal to another value. For example, in the inequality \(y \leq \sqrt{x+1}\), we are saying that the value of \(y\) can be equal to or less than the square root of \(x + 1\).
- The symbol \(\leq\) means "less than or equal to." It indicates that \(y\) can be equal to the square root, but also less.
- Inequalities help in defining regions in a graph, showing where a set of points lies in relation to a boundary.
Graphing Inequalities
Graphing inequalities involves plotting all points that satisfy a particular inequality on a cartesian coordinate system. To effectively graph \(y \leq \sqrt{x+1}\), follow these steps:
- First, consider the equation \(y = \sqrt{x+1}\) as the boundary. This will form the edge of the region you need to shade.
- Next, find points to plot the boundary curve. Use easy values like \(x = -1, 0,\) and \(3\) to get points \((x, y) = (-1, 0), (0, 1), (3, 2)\).
- Draw a curve that connects these points. This parabola opens to the right, forming the boundary for the inequality.
- To know where to shade, choose a test point not on the boundary, like \((0,0)\). Substitute it into \(y \leq \sqrt{x+1}\). If it holds true, shade the region including this point.
Exploring Square Roots
The square root function plays a key role in our inequality \(y \leq \sqrt{x+1}\). Understanding square roots is crucial for simplifying expressions and graphing them correctly.
- The square root of a number \(x\) is a value which, when multiplied by itself, gives \(x\). It is denoted by \(\sqrt{x}\).
- In our graph, \(y = \sqrt{x+1}\), the expression inside the square root, \(x+1\), must be non-negative since square roots of negative numbers are not real. This requirement leads to \(x \geq -1\).
- Square roots only produce non-negative outputs. That means on the graph, \(y\) is always zero or greater (\(y \geq 0\)).
