Q. 21 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 7–22, solve each inequality. 

$6(x^2-1)>5x$

Step-by-Step Solution

Answer

The solution set is $\left\{x:x<\frac{-2}{3}\mathrm{or} \mathrm{x}>\frac{3}{2}\right\}$.

Step 1. Given Information

The given inequality is $6(x^2-1)>5x$.

Step 2. Find the Intercepts and Vertex

• Express the inequality in standard form.

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

• Factor the function $f\left(x\right)=6x^2-5x-6$.

$f\left(x\right)=(2x-3)(3x+2)$

• So, the x-intercept of the function are $\left(\frac{3}{2},0\right), \left(-\frac{2}{3},0\right)$.
• The value of $f\left(0\right)=-6$.
• So, the y-intercept is $(0,-6)$.
• Find x -coordinate of the vertex by using the formula $x=\frac{-b}{2a}$.

$x=\frac{5}{12}$

• The value of $f\left(\frac{5}{12}\right)=\frac{-169}{24}$.
• So, the vertex of the parabola is $\left(\frac{5}{12},\frac{-169}{24}\right)$..

Step 3. Plot the Function

• Plot the curve of the function on the graph by using the obtained vertices and intercepts.

Step 4. Find the region above x- axis

• From the graph, it is observed that the curve is above x-axis when $x<\frac{-2}{3}\mathrm{or} \mathrm{x}>\frac{3}{2}$.
• So, the solution set is $\left\{x:x<\frac{-2}{3}\mathrm{or} \mathrm{x}>\frac{3}{2}\right\}$