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- 1. 1. Multiply the first fraction by , and multiply the second fraction by , to get: Since the two fractions now have the same denominator, the numerators can be combined: This fraction cannot be reduced any further. 2. The quotient of two real numbers with different signs is 3. Simplify the following expression: Move the variables with the negative exponents to the opposite side of the fraction bar, and change those exponents from negative to positive. This gives: Combine like terms in the numerator and the denominator to get: Squaring all of the terms in the numerator and the denominator then gives: 4. Graph the following equation: Substituting x = 0 into the equation gives:
- 2. The point (0, 2) will be one point on the graph of this line. Substituting x = 3 into the equation gives: The point (3, 4) will be another point on the line. Plot the two points and connect with a solid line. The graph looks like this: 5. What are the equation and slope of the y-axis? The y-axis is a vertical line, so its slope is All of the points on the y-axis have an x coordinate of 0, so the equation of the line is 6. Given f(x) = 2x – 8, find f(3). Substitute 3 in place of x to get: 7. Solve the following inequality. Give each result in set notation and graph it: Dividing through by 3 gives: Simplifying then gives: The solution in set notation is The graph looks like this: 8. Solve the following inequality. Write the solution in
- 3. interval notation and graph it. Convert the inequality into a compound inequality: Subtracting 4 from each part gives: Dividing through by 2 then gives: The solution in interval notation is The graph of the solution set looks like this: 9. Simplify the following product. Distributing the -2t3u term through the parentheses gives: 10. Simplify the following expression fully: Multiplying the numerator and the denominator by ab gives: Simplifying the terms then gives: 11. Solve the equation 7(x + 5) = x – 1 Distribute the coefficient of 7 on the left side: Subtract x from both sides: Subtract 35 from both sides: Divide both sides by 6: 12. Completely factor the following expression: 16m4 – 1.
- 4. This is the difference of two squares, a2 – b2, and can be factored as (a + b)(a – b). The last factor on the right side is another difference of two squares, and can also be factored: The complete factorization is then: 13. Write the numeral 0.0072 in scientific notation. Move the decimal three places to the right, so that it follows the first non-zero digit: Then, since the decimal point was moved three places to the right, add an exponent of -3: 14. Perform the indicated operation and simplify completely. The numerator of the first fraction can be factored as: The denominator of the second fraction can be factored as: Substituting these into the expression gives: Cancelling the (x – 5) terms from the numerator and the denominator gives: Cancelling a y from the numerator and the denominator gives: Finally, dividing 6 by 2 leaves: 15. Solve the following equation for r: d = rt Dividing both sides by t gives: 16. Solve the system of equations given below. 3x + y = 12 x – y – 2z = 10 2x + 3y + 5z = -7 Solving the first equation for y gives: Substituting this in place of y in the second equation gives:
- 5. Substituting for x and y in the third equation gives: Substituting for x again then gives: Expanding terms and simplifying gives: Multiplying through by 2 gives: With the value of z known, the value of x can be determined: Then, with the value of x known, the value of y can be determined: 17. Do the following two lines intersect? Answer yes or no, together with the point of intersection, if any. 5x + 6y = -5.5 6x + 1.5y = -8.5 Rearranging the first equation into y = mx + b form gives: Rearranging the second equation into the same form gives: The slope of the first line is -5/6. The slope of the second line is -4. Since the slopes of the two lines are different, the two lines will intersect at some point. Setting the right of each equation equal gives: Multiplying through by 12 then gives: Adding 11 to both sides gives: Substituting this into one of the two equations for y gives: 18. Compute the determinant: 19. Compute the distance between the two points and
- 6. 20. Rationalize the denominator of To rationalize the denominator, multiply both numerator and denominator by the conjugate of the denominator: Multiplying the fractions and simplifying gives: 21. The volume (V) of a cylinder with radius (r) and height (h) is given by V = πr2h. Solve this formula for r. x6 4y10 x 6 4y 10 y = 2 3 x + 2 y= 2 3 x+2 −3≤ 3x <12
- 7. -3£3x<12 2y + 4 <10 2y+4<10 −2t3u t0u4 − 4t2u3( ) -2t 3 ut 0 u 4 -4t 2 u 3 () 1 a + 1 b a b − b
- 8. a 1 a + 1 b a b - b a x2 + 2x − 35 y ∗ 6y3 2x −10 x 2 +2x-35 y * 6y 3 2x-10 3 0 −1 6 −1 4 2 −2 −2
- 9. 30-1 6-14 2-2-2 1− 2,−1( ) 1-2,-1 () 2+ 2,4( ) 2+2,4 () 5− x 5− y 5-x 5-y 8 9 − 3 5 = 8 9 - 3 5