Use the graph to find solutions.

#### (a)

$i) x=dfrac{1}{2}, 2,$or $dfrac{7}{2}$

$ii) x=-1$ or $2$

#### (b)

$i) dfrac{1}{2} < x dfrac{7}{2}$

$ii) -1 < x < 2$

#### (c)

$i) xleq dfrac{1}{2}$; $2leq xleq dfrac{7}{2}$

$ii) xleq -1$ or $xgeq 2$

#### (d)

$i) dfrac{1}{2}leq x leq 2$ or $xgeq dfrac{7}{2}$

$ii) -1leq x leq 2$

#### (a)

$3=2^{2x}$

Try $x=1:3=2^2$

$3=4$ To high

Try $x=0.5:3=2^1$

$3=2$ Too low

Try $0.6:3=2^{1.2}$

$3=2.3$ Too low

Try $0.8:3=2^{1.6}$

$3=3.03$

So, $x=0.8$

#### (b)

$0=sin(0.25x^2)$

Try $x=0$: $0=sin0$

$0=0$ Correct

Try $x=2$: $0=sin(0.25(4))$

$0=0.84$ Too high

Try $x=3$: $0=sin(0.25(9))$

$0=0.78$ Too high

Try $x=3.5$: $0=sin(0.25(12.25))$

$0=0.08$ Close

Try $x=3.6$: $0=sin(0.25(12.96))$

$0=-0.1$

So, $x=0$ and $3.5$

#### (c)

$3x=0.5x^3$

Try $x=-2: 3(-2)=0.5(-2)^3$

$-6=-4$

Try $x=-3: 3(-3)=0.5(-3)^3$

$-9=-13.5$

Try $x=-2.5: 3(-2.5)=0.5(-2.5)^3$

$-7.5=-7.8$

Try $x=-2.4: 3(-2.4)=0.5(-2.4)^3$

$-7.2=-6.9$

So, $x=-2.4$

#### (d)

$cos x=x$

Try $x=0$: $cos 0=0$

$1=0$

Try $0.5$ : $cos0.5=0.5$

$0.8=0.5$

Try $0.6$: $cos 0.6=0.6$

$0.8=0.6$

Try $0.7$: $cos0.7=0.7$

$0.76=0.7$

So, $x=0.7$

(a)$x=0.8$; (b) $x=0$ and $3.5$ (c) $x=-2.4$; (d) $x=0.7$

Graph and use the graph to find the solutions.

$x=-1.3$ or $1.8$

From the graph, we can see $textbf{following solutions:}$

$f(x)<g(x): 1.3< x g(x): 0 < x < 1.3$ or $1.6 < x < 3$

$f(x)<g(x): 1.3< x g(x): 0 < x < 1.3$ or $1.6 < x < 3$

#### (a)

$5sec{x}=-x^2$

We can try $x=2$: $5sec2=-2^2 Rightarrow -12=-4$

We can try $x=2.5$: $5sec2.5=-2.5^2 Rightarrow -6.24=-6.25$

$textbf{So}$, $x=2.5$
#### (b)

$sin^3{x}=sqrt{x}-1$

We can try $x=2$: $sin^32=sqrt{2}-1Rightarrow 0.75=0.41$

Now, we can try $x=2.2$: $sin^32.2=sqrt{2.2}-1 Rightarrow 0.53=0.48$

$textbf{So}$, $x=2.2$
#### (c)

$5^x=x^5$

We can try $x=1$: $5^1=1^5 Rightarrow 5=1$

Now, we can try $x=2$: $5^2=2^5 Rightarrow 25=32$

We can try $x=1.9$: $5^1.9=1.9^5 Rightarrow 21.28=24.76$

We can try $x=1.8$: $5^1.8=1.8^5 Rightarrow 18.12=18.9$

$textbf{So}$, $x=1.8$.

#### (d)

$cos{x}=dfrac{1}{x}$

We can try $x=-2$: $cos-2=dfrac{1}{-2} Rightarrow -0.42=-0.5$

We can try $x=-2.1$: $cos-2.1=dfrac{1}{-2.1} Rightarrow -0.5=-0.58$

$$
textbf{So, $x=-2.1$.}
$$

#### (e)

$log{x}=(x-10)^2+1$

We can try $x=9$: $log9=(9-10)^2+1 Rightarrow 0.95=0$

We can try $x=10$: $log10=(10-10)^2+1 Rightarrow 1=1$

$textbf{So, $x=10$}$.
#### (f)

$sin(2pi{x})=-4x^2+16x-12$

We can try $x=0$: $sin0=-4(0)^2+16cdot0-12 Rightarrow 0=-12$

We can try $x=1$: $sin1=-4(1)^2+16cdot1-12 Rightarrow 0=0$

We can try $x=3$: $sin6pi=-4(3)^2+16cdot3-12 Rightarrow 0=0$

$textbf{So, $x=1$ or $x=3$}$.

(a) $x=2.5$; (b) $x=2.2$; (c) $x=1.8$; (d) $x=-2.1$; (e) $x=10$; (f) $x=1$ or $x=3$

We can use $textbf{graphing calculator}$ to estimate the solutions.

#### (a)

$x=1.81$ or $x=0.48$

#### (b)

$x=-1.38$ or $x=1.6$
#### (c)

$x=-1.38$ or $x=1.30$
#### (d)

$x=-0.8$ or $x=0.8$
#### (e)

$x=0.21$ or $x=0.74$
#### (f)

$x=0$ or $x=0.18$ or $x=0.38$ or $x=1$

(a) $x=1.81$ or $x=0.48$; (b) $x=-1.38$ or $x=1.6$; (c) $x=-1.38$ or $x=1.30$; (d) $x=-0.8$ or $x=0.8$; (e) $x=0.21$ or $x=0.74$; (f) $x=0$ or $x=0.18$ or $x=0.38$ or $x=1$

Since the graph crosses the $x$-axis at $x=0.7$, the $x$-coordinate of the solution is $0.7$.Use $x=0.7$ to find the $y$-coordinate.

$y=-3x^2$

$y=-3(0.7)^2$

$y=-1.47$

So, the coordinates are $(0.7, -1.5)$.

$2.3(0.96)^t=1.95(0.97)^t$

We can use graphing calculator to estimate the solution, and according to it, we can conclude that $textbf{solution is}$ $t=15$.

So, they will be $textbf{about the same in}$ $1997+15$ or $2012.$

We can use $textbf{graphing calculator}$ to estimate the solutions.

$textbf{(a)}$ $xin(-0.57,1)$

$textbf{(b)}$ $xin[0,0.58]$

$textbf{(c)}$ $xin(-infty,0)$

$textbf{(d)}$ $xin(0.17,0.83)$

$textbf{(e)}$ $xin(0.35,1.51)$

$textbf{(f)}$ $xin(0.1,0.5)$

$textbf{(a)}$ $xin(-0.57,1)$

$textbf{(b)}$ $xin[0,0.58]$

$textbf{(c)}$ $xin(-infty,0)$

$textbf{(d)}$ $xin(0.17,0.83)$

$textbf{(e)}$ $xin(0.35,1.51)$

$textbf{(f)}$ $xin(0.1,0.5)$

Answers may vary. For example, $f(x)=x^3+5x^2+2x-8$ and $g(x)=0$

Answers may vary. For example, $f(x)=-x^2+25$ and $g(x)=-x+5$

$acos{x}=bx^3+6$

$acos(-1.2)=b(-1.2)^3+6$

$acos(-0.7)=b(-0.7)^3+6$

$0.36a=-1.728b+6$

$0.76a=-0.343b+6$

$dfrac{-1.728b+6}{0.36}=dfrac{-0.343b+6}{0.76}$

$-1.31328b+4.56=-0.12348b+2.16$

$-1.1898b=-2.4$

$b=2$

So, we have that $a=dfrac{-1.728cdot2+6}{0.36}=7$

$textbf{Finally, we have that}$ $a=7, b=2$.

$a=7, b=2$

We can use $textbf{graphing calculator}$ to determine the solutions.And, using it, we can conclude that:

$x=0pm2n, x=-0.67pm2n$, or $x=0.62pm2n, ninBbb{I}$.

$x=0pm2n, x=-0.67pm2n$, or $x=0.62pm2n, ninBbb{I}$.

We can use $textbf{graphing calculator}$ to determine the solutions, an we can conclude next using it:

$xin(2n,2n+1), ninBbb{I}$.

$xin(2n,2n+1), ninBbb{I}$.