Q. 20 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 7–22, solve each inequality.

$25x^2+16<40x$

Step-by-Step Solution

Answer

The inequality has no solution.

Step 1. Given information

The given inequality is $25x^2+16<40x$.

Step 2. Find the Intercepts and Vertex

• Express the inequality in standard form.

$25x^2-40x+16<0$

• Factor the function $f\left(x\right)=25x^2-40x+16$.

$f\left(x\right)={(5x-4)}^2$

• So, the x-intercept of the function is $\left(\frac{4}{5},0\right)$.
• The value of $f\left(0\right) = 16$.
• So, the y-intercept is $(0,16)$.
• Find x -coordinate of the vertex by using the formula$x=\frac{-b}{2a}$.

$\begin{array}{rcl}x& =& \frac{40}{50}\\ & =& \frac{4}{5}\end{array}$

• The value of $f\left(\frac{4}{5}\right)=0$.
• So, the vertex of the parabola is $\left(\frac{4}{5},0\right)$.

Step 3. Graph the function

• Plot $f\left(x\right)$ on a graph by using the intercepts and the vertex.

Step 4. Find the region below x -axis

The function $f\left(x\right)$ is on or above the horizontal axis, so the given inequality has no solution.