use analytical and graphical methods to solve the inequality.
x©ø+10x©÷+31 ¡ -30
(2x/x+1)¡(4/x-3)
To solve the inequality x² + 10x + 31 ≤ -30 using analytical and graphical methods, we can follow these steps:
1. Analytical Method:
a. Start by moving all terms to one side of the inequality to get the equation in standard quadratic form: x² + 10x + 31 + 30 ≤ 0.
b. Simplify the equation: x² + 10x + 61 ≤ 0.
c. To solve this quadratic inequality, we need to determine the values of x that make the inequality true. This can be done by analyzing the sign of the quadratic expression, x² + 10x + 61, for different values of x.
d. We observe that the quadratic expression is always positive (greater than 0) for all real numbers x. Therefore, there are no solutions for the inequality x² + 10x + 31 ≤ -30.
2. Graphical Method:
a. Plot the graph of the quadratic function y = x² + 10x + 31.
b. Identify the x-values for which the graph lies below the y = -30 line.
c. Shade the region where the graph of the quadratic function is below the y = -30 line.
d. By observing the graph, we can see that the graph of y = x² + 10x + 31 does not intersect the region below the y = -30 line, indicating that there are no solutions to the inequality x² + 10x + 31 ≤ -30.
Please note that the second inequality you provided, (2x/x+1)¡(4/x-3), seems to be incomplete or incorrectly formatted. Could you please provide the full and correct inequality so that I can help you solve it?