Q. 31 - Precalculus Enhanced with Graphing Utilities

Question

In parts (a) and (b), use the following figure. 

(a) Solve the equation: $F\left(x\right)=g\left(x\right)$.

(b) Solve the inequality: $g\left(x\right)\le f\left(x\right)<h\left(x\right)$

Step-by-Step Solution

Answer

Part (a) $x=-6$.

Part (b) $\left\{x|-6\le x<5\right\}$.

Part (a) Step 1. Given information.

The given figure is as follows,

There are three lines $f\left(x\right), g\left(x\right), h\left(x\right)$.

Part (a) Step 2. Explanation.

The equation $f\left(x\right)=g\left(x\right)$ represent the intersection of graph $f\left(x\right)$ and $g\left(x\right)$.

Here  intersection of graph $f\left(x\right), g\left(x\right)$ is $\left(-6,-5\right)$, the value of x is -6.

Hence solution of $f\left(x\right)=g\left(x\right)$ is $x=-6$.

Part (b) Step 1. Explanation.

The points $\left(0,12\right), \left(5,12\right)$ are on $h\left(x\right)$.

The points $\left(-6,-5\right), \left(0,-5\right)$ are on $g\left(x\right)$.

The points $\left(5,12\right), \left(-6,-5\right)$ are on $f\left(x\right)$.

For all values of x,  function h(x) is greater than g(x).

The function f(x) is intersecting h(x) at $(5,12)$ and $g\left(x\right)$ at $\left(-6,-5\right)$.

For $g\left(x\right)\le f\left(x\right)<h\left(x\right)$ , here lower value of x is -6 and upper value is 5.

So solution of inequality is $\left\{x|-6\le x<5\right\}$.