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.
Begin by replacing the variables in the expression with the given values. Then simplify the resulting expression paying careful attention to order of operations.
2x² + 4y
2(3)² + 4(5)
2(9) + 4(5)
18 + 20
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.
-½x + 8 > 14
-½x + 8 – 8 > 14 – 8
-½x > 6
(-2)(-½x) < -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.
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.
6x – 8y = 24
6x – 8(0) = 24
6x – 0 = 24
6x = 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.
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
x² – 7x = -12
x² – 7x + 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.
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 a² + log b³
2log a + 3log b
2x + 3y
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 email@example.com.