Mat QS
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mat_qs.docx
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:
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 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.
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:
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:
