Problem 58 - Precalculus : Building Concepts and Connections

Question

Use a graphing utility to find all real solutions. You may need to adjust the window size manually or use the ZOOMFIT feature to get a clear graph. $$\text { Solve } \sqrt{0.3 x+0.95}-\sqrt{0.75 x-0.5}=-0.3$$

Step-by-Step Solution

Answer

The real solutions can be identified from the x-values of the points where the graph intersects the x-axis.

Step 1: Rewrite the equation

First, rewrite the original equation \(\sqrt{0.3 x+0.95}-\sqrt{0.75 x-0.5}=-0.3\) in a suitable format to be graphed. So it should be written as \[f(x) = \sqrt{0.3x + 0.95} - \sqrt{0.75x - 0.5} + 0.3\]. Notice that the graph of this function has to cross the x-axis at all real solutions of the original equation.

Step 2: Graph the equation

Now using a graphing utility plot the function \(f(x)\). Be mindful in selecting a suitable window or zoom level to see when and where the graph cuts the x-axis.

Step 3: Identify the real solutions

Identify the x-values where the graph of the function intersects the x-axis, like moving the cursor to the point of intersection or check 'Zeroes' from the menu of the graphing utility, depending on the tool used. These x-values will be the solutions for the original equation.