Problem 124 - College Algebra

Question

Use a graphing utility and the graph's \(x\) -intercepts to solve each equation. Check by direct substitution. A viewing rectangle is given. $$\begin{aligned} &\sqrt{2 x+13}-x-5-0\\\ &[-5,5,1] \text { by }[-5,5,1] \end{aligned}$$

Step-by-Step Solution

Answer

The solutions are the x-values of the points where the graph of the equation intercepts the x-axis. These can be found using a graphing utility with the specified viewing rectangle. The exact values will depend on the graph but should be checked by substituting back into the original equation.

Step 1: Plotting the Equation

Use a graphing utility to plot the equation \( \sqrt{2x + 13} - x - 5 = 0 \) over the given viewing rectangle [-5,5,1] by [-5,5,1]. This rectangle indicates the ranges of x and y to plot the curve over, and our aim is to find where this curve crosses the x-axis (where y=0).

Step 2: Identifying the x-Intercepts

The points at which the curve intersects the x-axis are the solutions of the equation. These are points where y = 0. Identify these x-intercepts and note down the x-values of these points.

Step 3: Verifying by Substitution

To confirm that the identified x-intercepts are accurate, they should be substituted back into the original equation. If the left-hand side of the equation evaluates to 0 for these x-values, this means they are indeed solutions to the equation.

Key Concepts

Graphing Utility

A graphing utility is a tool, often a digital calculator or software, that helps us visualize equations.
To use it, we input the equation and specify a range, like a viewing rectangle, to see where the graph crosses certain lines.
In this problem, we are dealing with the equation \( \sqrt{2x + 13} - x - 5 = 0 \). Using a graphing utility can allow us to see where the graph crosses the x-axis.
This is crucial, as these crossing points are where the solution to our equation lies.

• Ensure you input the equation correctly into the utility.
• Make sure the settings are set according to the specified viewing rectangle.
• Observe carefully where the graph meets the x-axis, as these are your x-intercepts.

Graphing utilities help find solutions visually and are invaluable for solving equations when algebraic methods might be cumbersome.

X-Intercepts

X-intercepts are critical in solving equations graphically. They are the points where the graph of the equation crosses the x-axis.
At these points, the y-value is zero because the graph levels out and touches the axis.
For the equation \( \sqrt{2x + 13} - x - 5 = 0 \), once plotted using a graphing utility, any point on the x-axis where the graph meets represents an x-intercept.

• Observe the exact x-values at which the graph meets the x-axis.
• These x-values are solutions to the equation because this is where the equation equals zero.

Understanding x-intercepts is crucial because finding where the graph crosses the x-axis makes it easier to solve for these points using visual estimation or direct reading from the graph.

Direct Substitution

Direct substitution helps confirm the solutions found using graphing methods.
This involves substituting the x-intercepts back into the original equation.
For the equation \( \sqrt{2x + 13} - x - 5 = 0 \), substitute the x-values from identified intercepts into it.

• Calculate the left hand side of the equation using these x-values.
• If the result equals zero, the x-value is a correct solution.
• Through substitution, verify all intercepts to check accuracy.

Using direct substitution reinforces the results found using graphical methods, and is a reliable way to ensure that the correct solutions have been identified.

Viewing Rectangle

A viewing rectangle specifies the part of the graph you want to view when plotting an equation.
It's a defined range for the x and y values which frames the 'window' through which you view the graph.
In the given problem, the viewing rectangle is \([-5,5,1]\) by \([-5,5,1]\), aiming to contain and display the pertinent part of the graph.

• The first interval \([-5,5,1]\) represents the x-minimum, x-maximum and the x-step.
• The second, \([-5,5,1]\), does similarly for the y-axis.
• A well-chosen viewing rectangle ensures that the graph is displayed clearly and makes it easier to identify x-intercepts.

Understanding how to set and interpret the viewing rectangle is essential for accurate graphing and, consequently, for correctly solving equations using graphical methods.