Question

In problems 7-22, solve the inequality.

$6x^2<6+5x$

Short answer

The solution set is$(\frac{-2}{3},\frac{3}{2})$.

Step-by-step solution

Step 1. Given Information

The given inequality is$6x^2<6+5x$.

Step 2. Find the intercepts and vertex

• Rewrite the given inequality.

$6x^2-5x-6<0$

• The value of $f\left(0\right)=-6$.
• So, the y-intercept is $(0,-6)$.
• Factor the equation.

$\begin{array}{rcl}6x^2-5x-6& =& 0\\ 6x^2-9x+4x-6& =& 0\\ 3x(2x-3)+2(2x-3)& =& 0\\ (3x+2)(2x-3)& =& 0\end{array}$

• Equate the factors to 0.

$\begin{array}{rcl}3x+2& =& 0\\ x& =& \frac{-2}{3}\\ 2x-3& =& 0\\ x& =& \frac{3}{2}\end{array}$

• So, the x-intercepts are $(\frac{-2}{3},0),(\frac{3}{2},0)$.
• The vertex of the parabola exists at $\frac{-b}{2a}=\frac{-(-5)}{2\left(6\right)}=\frac{5}{12}$.
• Find the value of the function at $x=\frac{5}{12}$.

$\begin{array}{rcl}f\left(\frac{5}{12}\right)& =& 6{\left(\frac{5}{12}\right)}^2-5\left(\frac{5}{12}\right)-6\\ & =& 6\left(\frac{25}{144}\right)-\frac{25}{12}-6\\ & =& \frac{150-300-864}{144}\\ & =& \frac{1014}{144}\\ & =& \frac{169}{24}\end{array}$

• So, the vertex is $(\frac{5}{12},\begin{array}{rcl}& & \frac{169}{24}\end{array})$.

Step 3. Plot the graph

Plot the parabola on a graph using the vertex and intercepts calculated in the previous step.

Step 4. Find the solution set

• From the graph, it is observed that for x belonging to $(\frac{-2}{3},\frac{3}{2})$, $y<0$.
• So, the solution set of the given inequality is $(\frac{-2}{3},\frac{3}{2})$.