How can you use a graph to distinguish between a square root inequality that includes a "greater than" symbol and one that includes a "greater than or equal to" symbol?

Answer 1

When analyzing square root inequalities, the distinction between a "greater than" (>) and a "greater than or equal to" (≥) symbol can be visually represented through graphs. Here’s how you can use a graph to distinguish between these types of inequalities:

1. Understanding the Inequalities:

   • For an inequality like \( \sqrt{x} > a \), we are looking for the values of \( x \) where \( \sqrt{x} \) exceeds \( a \). This inequality does not include the point where \( \sqrt{x} = a \).
   • For an inequality like \( \sqrt{x} \geq a \), we want the values of \( x \) where \( \sqrt{x} \) is either greater than or equal to \( a \). This inequality includes the point where \( \sqrt{x} = a \).

2. Graphing the Function:

   • First, graph the function \( y = \sqrt{x} \). This is a curve that starts at the origin (0,0) and rises to the right, becoming less steep as \( x \) increases.
   • Next, identify the horizontal line \( y = a \) on the same graph. This line represents the constant value of \( a \).

3. Identify the Points of Intersection:

   • Determine where the curve \( y = \sqrt{x} \) intersects the line \( y = a \). This point is where \( x = a^2 \).

4. Analyzing the Regions:

   • For \( \sqrt{x} > a \):

     • The solution set includes values of \( x \) that lie in the region above the line \( y = a \) starting from \( x = a^2 \) onward (to the right).
     • At the point \( (a^2, a) \), the value \( \sqrt{x} \) equals \( a \), and this point is not included in the solution set. You can indicate this on the graph with an open dot at the point \( (a^2, a) \).

   • For \( \sqrt{x} \geq a \):

     • The solution set includes values of \( x \) that lie in the region above and on the line \( y = a \).
     • At the point \( (a^2, a) \), the value \( \sqrt{x} \) is equal to \( a \), and this point is included in the solution set. You can indicate this on the graph with a closed dot at the point \( (a^2, a) \).

5. Conclusion:

   • On the graph:
     • For \( \sqrt{x} > a \), you will have an open circle at \( (a^2, a) \) and shade the region above the line (representing \( x > a^2 \)).
     • For \( \sqrt{x} \geq a \), you will have a closed circle at \( (a^2, a) \) and shade the region above the line (including the point).

By following these steps, you can visually distinguish between the two types of square root inequalities.