Problem 67 - Algebra 2

Question

Simplify. Graph \(y \leq \sqrt{x+1}\)

Step-by-Step Solution

Answer

Graph the parabola and shade below it for \(x \geq -1\).

Step 1: 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 \(x + 1\).

Step 2: 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.

Step 3: 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, \(y \geq 0\).

Step 4: 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.

Step 5: 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.

Step 6: 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.

Key Concepts

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.

When working with inequalities, it's crucial to identify which area on the graph truly represents all the potential solutions. This region includes all the points that satisfy the inequality, helping us see the solutions visually.

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.

The shading represents all solution sets that fit the inequality criteria, providing a visual picture of all possible points that work within the given conditions.

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 (\(y \geq 0\)).

Understanding this behavior is important as it dictates the domain and range of the function you're working with. This further supports determining where on the graph you need to draw your curves and shade, ensuring you're accurately showing all potential solutions.