Hopefully you were able to find some time and a quiet place this weekend to try our Five Problems for Your Weekend (3/13/15). Below you’ll find the fully worked-out solutions to those problems.

11. C

Begin by replacing the variables in the expression with the given values. Then simplify the resulting expression paying careful attention to order of operations.

2*x**² + *4y

2(3)² + 4(5)

2(9) + 4(5)

18 + 20

38

38. H

You just need to carefully solve the inequality using opposite operations. In the final step there will be two important things that you need to be *very* careful with.

-½*x* + 8 > 14

-½*x* + 8 – 8 > 14 – 8

-½*x* > 6

(-2)(-½*x*) < -2(6)

*x* < – 12

Two important ideas from the last step:

- Dividing by a fraction is the same as multiplying by its reciprocal. So dividing by -½ is the same as multiplying by -2.
- When solving an inequality, if you
*multiply*or*divide*by a negative you must switch the direction of the inequality symbol.

35. D

When a line crosses the *x*-axis, the *y*-coordinate of that point (the *x*-intercept) is always zero. You can substitute 0 for *y* in the equation and then solve for *x*.

6*x* – 8*y* = 24

6*x* – 8(0) = 24

6*x* – 0 = 24

6*x* = 24

*x* = 4

The line crosses the *x*-axis at the point (4, 0), so the value of *a* is 4. Alternatively, you could have put the equation into slope-intercept form (*y* = 3/4 *x* – 3), graphed it on your calculator and then looked at where it crosses the *x*-axis.

48. J

The determinant of the matrix is given by (*x*)(*x*) – (7)(*x*) and we are told that this is equal to -12. Now you can write and solve an equation to find the value(s) of *x* for which this is true.

(*x*)(*x*) – (7)(*x*) = -12

*x*² – 7*x* = -12

*x*² – 7*x *+ 12 = 0

(*x* – 4)(*x* – 3) = 0

*x* = 4; *x* = 3

You also could have plugged each of the answer choices into the matrix and evaluated the determinant until you found one that gave you a determinant equal to -12. Be careful with answer choice F — even though 3 works, it might not be the *only* answer. You would need to try any other answer that also includes 3.

55. E

In order to get this problem correct, you need to be very familiar with the log rules (summarized in the table below).

Name of Rule | Rule |
---|---|

Product | log A·B = log A + log B |

Quotient | log (A/B) = log A – log B |

Power | log (A^B) = B·log A |

Start by simplifying log a²b³ using the Product and Power rules.

log a²b³

log a² + log b³

2log a + 3log b

2*x* + 3*y*

In the final step, recall that it was given that log a = *x* and log b = *y.*

If you have questions about these problems or anything else to do with the ACT, leave a comment below or send me an email at info@cardinalec.com.

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