Solve and graph on a number line. \( 18 x^{2}+9 x+10>-10 \)

First, let's simplify the inequality by bringing all terms to one side: \[ 18x^2 + 9x + 10 + 10 > 0 \] This simplifies to: \[ 18x^2 + 9x + 20 > 0 \] Now, to find where this quadratic expression is greater than zero, we can first determine the roots using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 18 \), \( b = 9 \), and \( c = 20 \). Calculating the discriminant: \[ b^2 - 4ac = 9^2 - 4(18)(20) = 81 - 1440 = -1359 \] Since the discriminant is negative, the quadratic has no real roots, meaning it does not cross the x-axis. To determine the sign of the quadratic expression, we can evaluate it at a test point, say, \( x=0 \): \[ 18(0)^2 + 9(0) + 20 = 20 > 0 \] Since the quadratic is positive at \( x=0 \) and has no real roots, it is positive for all real values of \( x \). We can graph this by drawing a number line and shading all of it to represent: \( (-\infty, +\infty) \). The solution is all real numbers. Keep in mind, quadratics that don’t touch the x-axis can be a bit tricky, like trying to dance on a slippery floor — you can slip and not even realize! Always check the sign of the quadratic at a test point for confirmation. Now, if you’re interested in exploring inequalities further, don't forget to check polynomial long division — it's like breaking down a cake for easier sharing! Or, delve into the realm of complex numbers, which adds an exciting twist to your math adventures!