Chapter 2

For the following exercises, use your graphing calculator to input the linear graphs in the \(\mathrm{Y}=\) graph menu. After graphing it, use the \(2^{\text { nd }}\) CALC button and l:value button, hit ENTER. At the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the \(x\)-intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the \(x\)-intercept, hit ENTER. Now it says “guess?” Move your cursor to the left somewhere in between the left and right bound near the \(x\)-intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the x-intercept or the “zero” to the \(y\)-value. Use this to find the \(x\)-intercept. Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact \(x\)-intercept between your right and left boundaries. With other types of functions (more than one \(x\)-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries. $$\mathrm{Y}_{1}=-8 x+6$$

Note: With linear/straight line functions the zero is not really a "guess," but it is necessary to enter a "guess" so it will search and find the exact \(x\) -intercept between your right and left boundaries. With other types of functions (more than one \(x\) -intercept), they may be irrational numbers so "guess" is more appropriate to give it the correct limits to find a very close approximation between the left nd right boundaries. $$ Y_{1}=-8 x+6 $$

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{(1+3 i)(2-4 i)}{(1+2 i)} $$

Abercrombie and Fitch stock had a price given as \(P=0.2 t^{2}-5.6 t+50.2,\) where \(t\) is the time in months from 1999 to \(2001 .(t=1 \text { is January } 1999) .\) Find the two months in which the price of the stock was \(\$ 30 .\)

For the following exercises, use your graphing calculator to input the linear graphs in the \(\mathrm{Y}=\) graph menu. After graphing it, use the \(2^{\text { nd }}\) CALC button and l:value button, hit ENTER. At the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the \(x\)-intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the \(x\)-intercept, hit ENTER. Now it says “guess?” Move your cursor to the left somewhere in between the left and right bound near the \(x\)-intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the x-intercept or the “zero” to the \(y\)-value. Use this to find the \(x\)-intercept. Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact \(x\)-intercept between your right and left boundaries. With other types of functions (more than one \(x\)-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries. $$\mathrm{Y}_{1}=4 x-7$$

Note: With linear/straight line functions the zero is not really a "guess," but it is necessary to enter a "guess" so it will search and find the exact \(x\) -intercept between your right and left boundaries. With other types of functions (more than one \(x\) -intercept), they may be irrational numbers so "guess" is more appropriate to give it the correct limits to find a very close approximation between the left nd right boundaries. $$ Y_{1}=4 x-7 $$

Given that the following coordinates are the vertices of a rectangle, prove that this truly is a rectangle by showing the slopes of the sides that meet are perpendicular. \((-1,1),(2,0),(3,3),\) and \((0,4)\)

Suppose that an equation is given \(p=-2 x^{2}+280 x-1000,\) where \(x\) represents the number of items sold at an auction and \(p\) is the profit made by the business that ran the auction. How many items sold would make this profit a maximum? Solve this by graphing the expression in your graphing utility and finding the maximum using \(2^{\text { nd }}\) CALC maximum. To obtain a good window for the curve, set \(x[0,200]\) and \(y[0,10000].\)

The formula for the circumference of a circle is \(C=2 \pi r\) . Find the circumference of a circle with a diameter of 12 in. (diameter \(=2 r ) .\) Use the symbol \(\pi\) in your final answer.

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{(3+i)^{2}}{(1+2 i)^{2}} $$

For the following exercises, use your graphing calculator to input the linear graphs in the \(\mathrm{Y}=\) graph menu. After graphing it, use the \(2^{\text { nd }}\) CALC button and l:value button, hit ENTER. At the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the \(x\)-intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the \(x\)-intercept, hit ENTER. Now it says “guess?” Move your cursor to the left somewhere in between the left and right bound near the \(x\)-intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the x-intercept or the “zero” to the \(y\)-value. Use this to find the \(x\)-intercept. Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact \(x\)-intercept between your right and left boundaries. With other types of functions (more than one \(x\)-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries. $$Y_{1}=\frac{3 x+5}{4} \text { Round your answer to the nearest thousandth.}$$

Note: With linear/straight line functions the zero is not really a "guess," but it is necessary to enter a "guess" so it will search and find the exact \(x\) -intercept between your right and left boundaries. With other types of functions (more than one \(x\) -intercept), they may be irrational numbers so "guess" is more appropriate to give it the correct limits to find a very close approximation between the left nd right boundaries. \(\mathrm{Y}_{1}=\frac{3 x+5}{4}\) Round your answer to the nearest thousandth.

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter \(\mathrm{Y} 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, liabs ( Find the points of intersection, recall \((2^{\text { nd }}\) CALC 5:intersection, lst curve, enter, } \(2^{\text { nd }}\) curve, enter, guess, enter). Copy a sketch of the graph and shade the \(x\) -axis for your solution set to the inequality. Write final answers in interval notation. $$ |x+2|-5 < 2 $$

Input the left-hand side of the inequality as a \(\mathrm{Y} 1\) graph in your graphing utility. Enter \(\mathrm{Y} 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, 1:abs(. Find the points of intersection, recall ( \(2^{\text {nd }}\) CALC 5:intersection, 1st curve, enter, \(2^{\text {nd }}\) curve, enter, guess, enter). Copy a sketch of the graph and shade the \(x\) -axis for your solution set to the inequality. Write final answers in interval notation $$ |x+2|-5<2 $$

A formula for the normal systolic blood pressure for a man age \(A\) , measured in mm Hg, is given as \(P=0.006 A^{2}-0.02 A+120 .\) Find the age to the nearest year of a man whose normal blood pressure measures 125 mm Hg.

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{3+2 i}{2+i}+(4+3 i) $$

A man drove 10 \(\mathrm{mi}\) directly east from his home, made a left turn at an intersection, and then traveled 5 \(\mathrm{mi}\) north to his place of work. If a road was made directly from his home to his place of work, what would its distance be to the nearest tenth of a mile?

A man drove 10 mi directly east from his home, made a left urn at an intersection, and then traveled 5 mi north to his place of work. If a road was made directly from his home to his place of work, what would its distance be to the nearest tenth of a mile?

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter \(\mathrm{Y} 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, liabs ( Find the points of intersection, recall \((2^{\text { nd }}\) CALC 5:intersection, lst curve, enter, } \(2^{\text { nd }}\) curve, enter, guess, enter). Copy a sketch of the graph and shade the \(x\) -axis for your solution set to the inequality. Write final answers in interval notation. $$ -\frac{1}{2}|x+2|<4 $$

Input the left-hand side of the inequality as a \(\mathrm{Y} 1\) graph in your graphing utility. Enter \(\mathrm{Y} 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, 1:abs(. Find the points of intersection, recall ( \(2^{\text {nd }}\) CALC 5:intersection, 1st curve, enter, \(2^{\text {nd }}\) curve, enter, guess, enter). Copy a sketch of the graph and shade the \(x\) -axis for your solution set to the inequality. Write final answers in interval notation. $$ -\frac{1}{2}|x+2|<4 $$

The slope for a wheelchair ramp for a home has to be \(\frac{1}{12} .\) If the vertical distance from the ground to the door bottom is 2.5 \(\mathrm{ft}\) , find the distance the ramp has to extend from the home in order to comply with the needed slope.

The cost function for a certain company is \(C=60 x+300\) and the revenue is given by \(R=100 x-0.5 x^{2} .\) Recall that profit is revenue minus cost. Set up a quadratic equation and find two values of \(x\) (production level) that will create a profit of \(\$ 300\) .

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{4+i}{i}+\frac{3-4 i}{1-i} $$

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter \(\mathrm{Y} 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, liabs ( Find the points of intersection, recall \((2^{\text { nd }}\) CALC 5:intersection, lst curve, enter, } \(2^{\text { nd }}\) curve, enter, guess, enter). Copy a sketch of the graph and shade the \(x\) -axis for your solution set to the inequality. Write final answers in interval notation. $$ |4 x+1|-3>2 $$

Input the left-hand side of the inequality as a \(\mathrm{Y} 1\) graph in your graphing utility. Enter \(\mathrm{Y} 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, 1:abs(. Find the points of intersection, recall ( \(2^{\text {nd }}\) CALC 5:intersection, 1st curve, enter, \(2^{\text {nd }}\) curve, enter, guess, enter). Copy a sketch of the graph and shade the \(x\) -axis for your solution set to the inequality. Write final answers in interval notation. $$ |4 x+1|-3>2 $$

If the profit equation for a small business selling \(x\) number of item one and \(y\) number of item two is \(p=3 x+4 y,\) find the \(y\) value when \(p=\$ 453\) and \(x=75\)

A falling object travels a distance given by the formula \(d=5 t+16 t^{2} \mathrm{ft}\) , where \(t\) is measured in seconds. How long will it take for the object to traveled 74 \(\mathrm{ft}\) ?

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{3+2 i}{1+2 i}-\frac{2-3 i}{3+i} $$

Given these four points: \(A(1,3), B(-3,5), C(4,7)\) and \(D(5,-4),\) find the coordinates of the midpoint of line segments \(\overline{A B}\) and \(\overline{C D}\) .

Given these four points: \(A(1,3), B(-3,5), C(4,7)\) and \(D(5,-4),\) fi \(\mathrm{d}\) the coordinates of the midpoint of line segments \(\overline{A B}\) and \(\overline{C D}\).

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter \(\mathrm{Y} 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, liabs ( Find the points of intersection, recall \((2^{\text { nd }}\) CALC 5:intersection, lst curve, enter, } \(2^{\text { nd }}\) curve, enter, guess, enter). Copy a sketch of the graph and shade the \(x\) -axis for your solution set to the inequality. Write final answers in interval notation. $$ |x-4|<3 $$

Input the left-hand side of the inequality as a \(\mathrm{Y} 1\) graph in your graphing utility. Enter \(\mathrm{Y} 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, 1:abs(. Find the points of intersection, recall ( \(2^{\text {nd }}\) CALC 5:intersection, 1st curve, enter, \(2^{\text {nd }}\) curve, enter, guess, enter). Copy a sketch of the graph and shade the \(x\) -axis for your solution set to the inequality. Write final answers in interval notation. $$ |x-4|<3 $$

A vacant lot is being converted into a community garden. The garden and the walkway around its perimeter have an area of 378 \(\mathrm{ft}^{2} .\) Find the width of the walkway if the garden is 12 \(\mathrm{ft}\) . wide by 15 \(\mathrm{ft}\) long.

For the following exercises, use this scenario: The cost of renting a car is \(\$ 45 / \mathrm{wk}\) plus \(\$ 0.25 / \mathrm{mi}\) traveled during that week. An equation to represent the cost would be \(y=45+0.25 x\), where \(x\) is the number of miles traveled. What is your cost if you travel \(50 \mathrm{mi}\) ?

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter \(\mathrm{Y} 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, liabs ( Find the points of intersection, recall \((2^{\text { nd }}\) CALC 5:intersection, lst curve, enter, } \(2^{\text { nd }}\) curve, enter, guess, enter). Copy a sketch of the graph and shade the \(x\) -axis for your solution set to the inequality. Write final answers in interval notation. $$ |x+2| \geq 5 $$

Input the left-hand side of the inequality as a \(\mathrm{Y} 1\) graph in your graphing utility. Enter \(\mathrm{Y} 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, 1:abs(. Find the points of intersection, recall ( \(2^{\text {nd }}\) CALC 5:intersection, 1st curve, enter, \(2^{\text {nd }}\) curve, enter, guess, enter). Copy a sketch of the graph and shade the \(x\) -axis for your solution set to the inequality. Write final answers in interval notation. $$ |x+2| \geq 5 $$

An epidemiological study of the spread of a certain influenza strain that hit a small school population found that the total number of students, \(P,\) who contracted the flu \(t\) days after it broke out is given by the model \(P=-t^{2}+13 t+130,\) where \(1 \leq t \leq 6\) . Find the day that 160 students had the flu. Recall that the restriction on \(t\) is at most \(6 .\)

For the following exercises, use this scenario: The cost of renting a car is \(\$ 45 / \mathrm{wk}\) plus \(\$ 0.25 / \mathrm{mi}\) traveled during that week. An equation to represent the cost would be \(y=45+0.25 x\), where \(x\) is the number of miles traveled. If your cost were \(\$ 63.75\), how many miles were you charged for traveling?

Solve \(|3 x+1|=|2 x+3|\)

For the following exercises, use this scenario: The cost of renting a car is \(\$ 45 / \mathrm{wk}\) plus \(\$ 0.25 / \mathrm{mi}\) traveled during that week. An equation to represent the cost would be \(y=45+0.25 x\), where \(x\) is the number of miles traveled. Suppose you have a maximum of \(\$ 100\) to spend for the car rental. What would be the maximum number of miles you could travel?

Solve \(x^{2}-x>12\)

The coordinates on a map for San Francisco are \((53,17)\) and those for Sacramento are \((123,78)\) . Note that coordinates represent miles. Find the distance between the cities to the nearest mile.

\(\frac{x-5}{x+7} \leq 0, x \neq-7\)

\(p=-x^{2}+130 x-3,000\) is a profit formula for a small business. Find the set of \(x\) -values that will keep this profit positive.

A small craft in Lake Ontario sends out a distress signal. The coordinates of the boat in trouble were (49, 64). One rescue boat is at the coordinates (60, 82) and a second Coast Guard craft is at coordinates (58, 47). Assuming both rescue craft travel at the same rate, which one would get to the distressed boat the fastest?

A small craft in ake Ontario sends out a distress signal. The coordinates of the boat in trouble were \((49,64) .\) One rescue boat is at the coordinates (60,82) and a second Coast Guard craft \(\mathrm{s}\) at coordinates (58,47) . Assuming both rescue craft travel at the same rate, which one would get to the distressed boat the fastest?

In chemistry the volume for a certain gas is given by \(V=20 T\) , where \(V\) is measured in \(c \mathrm{c}\) and \(T\) is temperature in \(^{\circ} \mathrm{C}\) . If the temperature varies between \(80^{\circ} \mathrm{C}\) and \(120^{\circ} \mathrm{C}\) , find the set of volume values.

A man on the top of a building wants to have a guy wire extend to a point on the ground 20 ft from the building. To the nearest foot, how long will the wire have to be if the building is 50 ft tall?

A basic cellular package costs \(\$ 20 / \mathrm{mo}\) . for 60 \(\mathrm{min}\) of calling, with an additional charge of \(\$ 0.30 / \mathrm{min}\) beyond that time. The cost formula would be \(C=\$ 20+.30(x-60) .\) If you have to keep your bill lower than \(\$ 50\) , what is the maximum calling minutes you can use?

If we rent a truck and pay a \(\$ 75 /\) day fee plus \(\$ .20\) for every mile we travel, write a linear equation that would express the total cost \(y,\) using \(x\) to represent the number of miles we travel. Graph this function on your graphing calculator and find the total cost for one day if we travel 70 \(\mathrm{mi}\) .