Question

Use a graphing calculator or a computer to show that the radical equation \(\sqrt{11 x-30}=-x\) has no solution. Explain.

Short answer

The equation \(\sqrt{11 x-30}=-x\) has no solutions because the left side of the equation (which is the output of a square root function) cannot be negative, whereas the right side of the equation is negative whenever x is positive.

Step-by-step solution

Analyze the radical equation

In any radical equation, the output of the square root function is always a non-negative number. This is because the square of any number, whether positive or negative, is always a positive number. This implies that the output of the square root function cannot be negative. Hence, \(\sqrt{11 x-30}\) can never be a negative number.

Observe right side of the equation

Examine the right side of the radical equation, which is -x. For this value to match the radical side, it needs to be non-negative (i.e., zero or positive). However, -x represents a negative value when x is positive.

Conclude the solution

From Steps 1 and 2, it is clear that a non-negative number (the square root side) can never equal a negative number (-x). Hence, the equation \(\sqrt{11 x-30}=-x\) does not have any solutions.