Q: Solve for $$x in $$the equation $$\dfrac{4}{x + 2} + \dfrac{3}{x - 2} = \dfrac{12}{(x + 2)(x - 2)}. $$Also, state any restrictions on $$x.$$

$$x \neq -2, 2; \varnothing (no $$solution)

Question 1

Solve the linear equation: $$(-7x + 3) + 5 = -6(x - 7). $$What is the value of $$x$$?

Accepted Answer

$$x = -34$$

Question 2

Solve for $$x in $$the equation $$\dfrac{4}{x + 2} + \dfrac{3}{x - 2} = \dfrac{12}{(x + 2)(x - 2)}. $$Also, state any restrictions on $$x.$$

Accepted Answer

$$x 
eq -2, 2; \varnothing (no $$solution)

Question 3

The U.S. Maritime Administration estimated the cost per ton of building an oil tanker as $$y = \dfrac{119,000}{x} + 220$$, where $$y is $$the cost in dollars per ton and $$x is $$the tons (in thousands). What size of oil tanker (in thousands of tons) can be built for $200 per ton?

Accepted Answer

$$375$$ thousand tons

Question 4

When 5 times a number is subtracted from 7 times the number, the result is 16. What is the number?

Accepted Answer

$$8$$

Question 5

You inherit $$42,000 and must invest it in two stocks paying 4% and 10% annual interest, respectively. If the total interest earned for the year is to be 2$$,400, how much should be invested at each rate?

Accepted Answer

$$30,000 at$$ $$4\%$$; $$12,000 at$$ $$10\%$$

Question 6

A rectangular room has a length that is 9 feet longer than twice its width. If the perimeter is 198 feet, what are the room's dimensions?

Accepted Answer

Width $$= 30 ft$$; Length $$= 69 ft$$

Question 7

The sum of the angles of a triangle is $$180^\circ. If $$one angle is four times the smallest angle and the third angle is $$30^\circ$$ greater than the smallest angle, what are the three angles?

Accepted Answer

$$25^\circ, 100^\circ, 55^\circ$$

Question 8

Solve the formula $$F = \dfrac{9}{5}C + 32$$ for $$C.$$

Accepted Answer

$$C = \dfrac{5}{9}(F - 32)$$

Question 9

Find the product and write the result in standard form: $$(8 - 9i)(-2 - 3i).$$

Accepted Answer

$$-43 - 6i$$

Question 10

Divide and express the result in standard form: $$\dfrac{3 - 2i}{2 + 3i}.$$

Accepted Answer

$$-i$$

Question 11

Solve the equation by the square root property: $$4(x - 4)^2$ = 8.$$

Accepted Answer

$$x = 4 \pm \sqrt{2}$$

Question 12

Solve the quadratic equation: $$5x^2 + 20x + 100 = 0$.

Accepted Answer

$$x = -2 \pm 4i$$

Question 13

A species of fish in a pond is modeled by $$P = 0.63x^2$ - 0.048x + 3$$, where $$P is $$the population in thousands and $$x is $$years after 1997. In what year will the population reach $$65,520$$ fish?

Accepted Answer

2007

Question 14

Solve and check the radical equation: $$24x + 48 = x + 8.$$

Accepted Answer

$$x = 4$$

Question 15

Solve and check the equation: $$(x + 6$$)^{3/2}$$ = 125.$$

Accepted Answer

$$x = 19$$

Question 16

Solve the equation by making an appropriate substitution: $$x^4$$ - $$2x^2 - 63 = 0$.

Accepted Answer

$$x = -3, 3, -i\sqrt{7}, i\sqrt{7}$$

Question 17

A formula for the velocity $$v (in ft/s) of an $$object falling from height $$h (in ft) is$$ $$v = \sqrt{2gh}$$, where $$g = 32 ft/s$^2. $$What is the velocity of a bowling ball after falling $$30$$ feet? (Round to the nearest tenth.)

Accepted Answer

$$43.8 ft/s$$

Question 18

Two cars leave an intersection. One travels north for $$18$$ miles, the other east. The distance between the cars is $$6$$ miles more than the distance traveled by the eastbound car. How far had the eastbound car traveled?

Accepted Answer

$$24$$ miles

Question 19

Claire has received scores of $$85$$, $$88$$, $$87$$, and $$75 on $$her algebra tests. What score must she receive on the fifth test to have an average of at least $$83$$?

Accepted Answer

$$80 or $$greater

Question 20

Solve the compound inequality and express the solution in interval notation: $$-4 \leq -4x - 12 \u003c 4.$$

Accepted Answer

$$(-4, -2]$$

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