Question

In Problems 7–22, solve each inequality.

$4x^2+9<6x$

Short answer

The given inequality has no solution.

Step-by-step solution

Step 1. Given Information

The given inequality is$4x^2+9<6x$.

Step 2. Find the Intercepts

• Rewrite the inequality in standard form.

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

• There are no real roots of $4x^2+9-6x=0$, 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 - intercept 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. Plot the Graph

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

Step 4. Find the region below x-axis

• From the graph, there is no region that lies below the x-axis.
• So, the given inequality has no solution.