Math 30+ Review #4 Solutions for selected exercises
Instructions for the exam
Show all work for full credit. Clearly circle or box your answers when applicable. When
ounding final answers, use three decimal places (unless there are fewer than 3, or unless 3
decimals is inappropriate for the context of the problem). You may use a pencil, eraser, ruler
(or other straight edge), scientific calculator, StatCrunch, the e-text, and desmos.com.
PS 8.1, 8.2, and 8.3
3) Solve
??âˆ’?
?
=
??+?
?
Lcd is 6: ? (
??âˆ’?
?
) = ? (
??+?
?
)
?(?? âˆ’ ?) = ?(?? + ?)
?? âˆ’ ? = ?? + ?
?? âˆ’ ?? = ? + ?
? = ?
Check:
?(?)âˆ’?
?
=
?(?)+?
?
?
â€¦
5) The sales of musical instruments were $?. ?? million in 2010, and they decreased by
about $?. ?? million per year until 2014.
a) Find an equation of a linear model that describes the situation. Explain what the
variables represent. (Hint: one of the variables will have to represent the number of
years since 2010).
Let ? = number of years since 2010, and let ?(?) = sales ? years after 2010, then
?(?) = ?? + ?
? = 0 ïƒ year 2010, when sales = $6.39 million ïƒ y-intercept = (0, 6.39) ïƒ b = 6.39
Decrease of $0.28 million ïƒ r.o.c. = m = -0.28
Equation: ?(?) = âˆ’?. ??? + ?. ??, in millions of dollars
) Predict the amount of sales in 2017.
2017 ïƒ ? = ? ïƒ ?(?) = âˆ’?. ??(?) + ?. ?? = ?. ?? ïƒ sales were $4.43 million in 2017
c) Estimate in which year sales were $?. ? million.
?. ? = âˆ’?. ??? + ?. ??
âˆ’?. ?? = âˆ’?. ???
?. ?? â€¦ = ?
So ? â‰ˆ ?. ?
? â‰ˆ ?. ? years since 2010 puts it right at the end of 2013, but this still means it hits the
sales of $5.3 million in the year 2013.
IA 1.3
6) Suppose that the percentages of men and women, ?(?) and ?(?) respectively, who have
completed for or more years of college are modeled by the system
? = ?(?) = ?. ??? + ??. ??
? = ?(?) = ?. ??? + ??. ??
where ? is the number of years since 1990.
a) Use substitution or elimination to estimate when the percentage of women who
have completed four or more years of college was equal to the percentage of men
who had completed four or more years of college.
Ie: find ? such that the expressions ? = ?(?) and ? = ?(?) are equal. To do this,
substitute the expression for ?(?) in for ? in the equation ? = ?(?):
?(?) = ?(?)
Then substitute in the respective equations:
?. ??? + ??. ?? = ?. ??? + ??. ??
And solve for ?:
?. ??? âˆ’ ?. ??? = ??. ?? âˆ’ ??. ??
?. ??? = ?. ??
? = ??. ?? â€¦ â‰ˆ ??. ?
About 22.9 years after 1990 is towards the end of 2012. So the percentage of women who
completed least 4 years of college equaled the percentage of men who completed at least 4
years of college somewhere near the end of 2012.
) What was that percentage?
?(??. ?) = ?(??. ?) = ?. ??(??. ?) + ??. ?? = ??. ?
So about 31.4% of both men and women had completed at least 4 years of college by the
end of 2012.
PS 8.4 & IA 3.5 (IA 3.5 was abso
ed into PS 8.4 lecture)
7) Solve the following formulas for the specified variable.
a) ?(? ?? ?) = ?(?) + ?(?) âˆ’ ?(? ??? ?); for ?(?)
?(? ?? ?) âˆ’ ?(?) + ?(? ??? ?) = ?(?) + ?(?) âˆ’ ?(?) âˆ’ ?(? ??? ?) + ?(? ??? ?)
?(? ?? ?) âˆ’ ?(?) + ?(? ??? ?) = ?(?)
) ?? =
??
âˆš?âˆ’?
; for ?
?? =
??
âˆš?âˆ’?
ïƒ (âˆš? âˆ’ ?) âˆ™ ?? = (âˆš? âˆ’ ?) âˆ™
??
âˆš?âˆ’?
ïƒ (âˆš? âˆ’ ?) âˆ™ ?? = ?? ïƒ âˆš? âˆ’ ? =
??
??
ïƒ (âˆš? âˆ’ ?)
?
= (
??
??
)
?
ïƒ ? âˆ’ ? = (
??
??
)
?
ïƒ ? = (
??
??
)
?
+ ?
PS 8.5
9) Solve the compound inequality ? â‰¤ ?(? âˆ’ ?) â‰¤ ?. Describe the solution set as an
inequality, in interval notation, and on a graph.
? â‰¤ ?(? âˆ’ ?) â‰¤ ?
? â‰¤ ? âˆ’ ?? â‰¤ ?
âˆ’? â‰¤ âˆ’?? â‰¤ âˆ’?
âˆ’?
âˆ’?
â‰¥ ? â‰¥
âˆ’?
âˆ’?
Inequality:
?
?
â‰¥ ? â‰¥
?
?
or, equaivalently,
?
?
â‰¤ ? â‰¤
?
?
Interval: [
?
?
,
?
?
]
Graph:
IA 4.1
10) Solve ? âˆ’ |??? +
?
?
| = âˆ’?
? âˆ’ |??? +
?
?
| = âˆ’?
âˆ’ |??? +
?
?
| = âˆ’?
|??? +
?
?
| = ?
â€¦so ??? +
?
?
, whatever it is, is a distance of 6 away from 0 on the number line. Only two
numbers are 6 away from 0 on the number line, namely -6 and 6. Soâ€¦
??? +
?
?
= âˆ’?
??? + ? = âˆ’??
??? = âˆ’??
? = âˆ’
??
??
Or
Or
??? +
?
?
= ?
??? + ? = ??
??? = ?
? =
?
??
=
?
?
Check by plugging both solutions back in to the original equation or by graphing.
13) Solve |?? + ?| âˆ’ ? < ?. Describe the solution set as an inequality, in interval notation,
and on a graph.
|?? + ?| âˆ’ ? < ?
|?? + ?| < ??
Whatever ?? + ? is, its distance from 0 along the number line is less than 12, so ?? + ? can
e anything between -12 and 12, exclusive. Soâ€¦
âˆ’?? < ?? + ? < ??
âˆ’?? < ?? < ??
âˆ’
??
?
?
??
?
Inequality: âˆ’
??
?
? < ?
Interval: (âˆ’
??
?
, ?)
Graph:
IA 1.6 (no systems)
17) Graph the linear inequality in two variables by hand: ?(? âˆ’ ?) + ? â‰¤ âˆ’?
Solve the inequality for ?: ?(? âˆ’ ?) + ? â‰¤ âˆ’?
?? âˆ’ ? + ? â‰¤ âˆ’?
? â‰¤ âˆ’?? + ?
So the equation for the boundary line is ? = âˆ’?? + ?
and, since the inequality â‰¤ allows for equality, we will
graph it with a solid line.
Its slope is ? = âˆ’? =
????
???
=
âˆ’?
?
, and its y-intercept is
(?, ?).
Since the point (?, ?) is easy to calculate with and it is
not on the boundary line, we can use it as a test point:
?((?) âˆ’ ?) + (?) â‰¤ âˆ’? ?
?(âˆ’?) â‰¤ âˆ’? ?
âˆ’? â‰¤ âˆ’? ?... yes
So (?, ?) is a solution to ?(? âˆ’ ?) + ? â‰¤ âˆ’?, and so is
every else on the same side of the boundary line as
(?, ?). So we indicate this by shading everything on
that side of the boundary line. Donâ€™t forget to label
the equation of the boundary line and the region of
solutions.
PS 9.1
19) a) Find the slope of the line that passes through the points (âˆ’?, ?) and (?, ?).
????? = ? =
?? âˆ’ ??
?? âˆ’ ??
=
? âˆ’ ?
? âˆ’ (âˆ’?)
= âˆ’
?
?
) Produce an equation in slope-intercept form for the line that passes through these
two points. (Verify that your equation is co
ect by graphing it in desmos.com)
S-I form: ? = ?? + ?
Substitute the slope in for ?: ? = âˆ’
?
?
? + ? and then sub in one of the ordered pairs to
find ?:
(?) = âˆ’
?
?
(?) + ?
? = âˆ’
?
?
+ ?
? +
?
?
= ?
??
?
= ?
So ? = âˆ’
?
?
? +
??
?
c)