Q. 22 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 7–22, solve each inequality. 

$2(2x^2-3x)>-9$

Step-by-Step Solution

Answer

The inequality is true on the entire real domain.

Step 1. Given Information

The given inequality is $2(2x^2-3x)>-9$.

Step 2. Find the intercepts and vertex

• Rewrite the inequality in standard form.

$4x^2-6x+9>0$

• There are no real roots of $4x^2-6x+9>0$, so, there are no x-intercepts of the curve.
• The value of $f\left(0\right) = 9$.
• So, the y-intercept is $(0,9)$.
• Find $x=\frac{-b}{2a}$to find the x - coordinate of the vertex of parabola.

$x=\frac{6}{8}$

• The value of $f\left(\frac{6}{8}\right)=\frac{27}{4}$ .
• So, the vertex is $\left(\frac{6}{8},\frac{27}{4}\right)$.

Step 3. Graph the Parabola

• Plot the parabola on a graph by using the obtained intercepts and the vertex.

Step 4. Find the region above the x- axis

From the graph, it is observed that the entire curve lies above the x-axis.

So, the entire domain of the function is the solution set.