How do I use a sign chart to solve 4x^2<=28x?

Oct 21, 2014

By subtracting $28 x$,

$4 {x}^{2} \le 28 x R i g h t a r r o w 4 {x}^{2} - 28 x \le 0$

Let $f \left(x\right) = 4 {x}^{2} - 28 x$.

First, turn it into an equation.

$R i g h t a r r o w 4 {x}^{2} - 28 x = 0$

by factoring out $4 x$,

$R i g h t a r r o w 4 x \left(x - 7\right) = 0 R i g h t a r r o w x = 0 , 7$

Use $0$ and $7$ to split the real number line into three intervals

$\left(- \infty , 0\right)$, $\left(0 , 7\right)$, and $\left(7 , \infty\right)$.

Choose any number from each interval as a sample point.
I choose $x = - 1 , 1 , 8$

$f \left(- 1\right) = 4 {\left(- 1\right)}^{2} - 28 \left(- 1\right) = 32 > 0$

$f \left(1\right) = 4 {\left(1\right)}^{2} - 28 \left(1\right) = - 24 < 0$

$f \left(8\right) = 4 {\left(8\right)}^{2} - 28 \left(8\right) = 32 > 0$

The above indicates that $f \left(x\right) < 0$ on $\left(0 , 7\right)$, which means that $f \left(x\right) \le 0$ on $\left[0 , 7\right]$ since $f \left(0\right) = f \left(7\right) = 0$.

Hence, the solution set of the inequality is $\left[0 , 7\right]$.

I hope that this was helpful.