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Q. 21

Expert-verified
Found in: Page 171

Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

In Problems 7–22, solve each inequality. $6\left({x}^{2}-1\right)>5x$

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

See the step by step solution

Step 1. Given Information

The given inequality is $6\left({x}^{2}-1\right)>5x$.

Step 2. Find the Intercepts and Vertex

• Express the inequality in standard form.

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

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

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

• 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 $\left(0,-6\right)$.
• 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 $\left\{x:x<\frac{-2}{3}\mathrm{or}\mathrm{x}>\frac{3}{2}\right\}$