Dealing with rational inequalities is like dealing with inequalities that feature rational expressions - these are expressions that take the form of a fraction in which both the numerator and the denominator are polynomials. The challenge when solving these inequalities lies in pinpointing the variable values that satisfy the inequality. It's a fascinating blend of concepts drawn from both inequality and polynomial equations.
| Topic | Problem | Solution |
|---|---|---|
| None | For the function $h(x)=\frac{6 x}{(x+1)(x-2)}$, s… | The inequality is \(h(x)<0\), where \(h(x)=\frac{6 x}{(x+1)(x-2)}\). |
| None | Solve the following inequality. \[ \frac{x}{6}>-7… | Translate the inequality \(\frac{x}{6}>-7\) into English: x divided by 6 is greater than -7. |
| None | 24) $\frac{3 x+2}{10}-\frac{1+6 x}{5} \leq-\frac{… | Rewrite the inequality as \(\frac{3 x+2}{10}-\frac{2(1+6 x)}{10} \leq -\frac{1}{2}\) |
| None | List the critical values of the related function.… | Identify the critical values of the function. The critical values are the values of x that make the… |
| None | List the critical values of the related function.… | The critical values of a function are the values of x that make the function equal to zero or undef… |
| None | For the function $h(x)=\frac{7 x}{(x+9)(x-2)}$, s… | First, we need to build a sign chart for the given expression \(\frac{7x}{(x+9)(x-2)}\). |
| None | List the critical values of the related function.… | First, identify the critical values of the function. The critical values are the values of x that m… |
| None | List the critical values of the related function.… | The critical values of a function are the x-values where the function is either undefined or its de… |
| None | List the critical values of the related function.… | Given the inequality \(\frac{x - 5}{x + 3} - \frac{x + 4}{x - 2} \leq 0\) |
| None | Solve the following inequality. \[ \sqrt[3]{-x^{2… | First, cube both sides of the inequality to remove the cube root. This gives us \(-x^{2}-1 > -343\). |
| None | Solve the rational inequality. Express your answe… | First, we subtract \(\frac{2x}{x+3}\) from both sides of the inequality to get \(\frac{3}{x+3} - \f… |
| None | $\sqrt{3 x-2<x}$ | First, we need to ensure that both sides of the inequality are defined and non-negative. Since the … |