Solutions
Name __________________________
Block ____ Date ___________
Algebra: 9.4.1: Systems of Inequalities
Bell Work: Write each equation in vertex form. Determine the maximum or minimum value.
a.
𝑦 = 𝑥 2 + 2𝑥 + 7
+1
𝑥
𝑥
1
𝑥
+1
𝑥2
𝑥
b.
-1
+7
𝒚 = (𝒙 + 𝟏)𝟐 + 𝟔
The minimum Value is 6
c. 𝑦 = 𝑥 2 − 4𝑥 − 5
𝑦 = 𝑥 2 − 8𝑥 − 5
−4 −4𝑥
𝑥
𝑥
2
𝑥
16
−2 −2𝑥
-16
−4𝑥 -5
𝑥
−4
𝒚 = (𝒙 − 𝟒)𝟐 − 𝟐𝟐
𝑥
4
-4
−2𝑥 -5
2
𝑥
−2
𝒚 = (𝒙 − 𝟐)𝟐 − 𝟗
The minimum Value is -21
The minimum Value is -9
9-89. Graph the system of inequalities:
y
shade solutions common to both
8
𝑦 ≤ −𝑥 + 3
6
2
𝑦> 𝑥−1
3
4
2
x
−8
−6
−4
−2
2
4
6
8
−2
−4
−6
−8
9-90. Graph the system of inequalities:
y
8
6
𝑦<𝑥+2
3
4
𝑦 ≤ 10 − 𝑥
4
2
x
−8
−6
−4
−2
2
−2
−4
−6
−8
4
6
8
9-91. Graph the system of inequalities:
𝑦 ≤ 𝑥2 + 𝑥 − 6
2
𝑦> 𝑥
3
9-92. Graph each system of inequalities:
a.
𝑦 ≥ 𝑥2 + 𝑥 − 6
b.
2
𝑦> 𝑥
3
𝑦 ≥ 𝑥2 + 𝑥 − 6
2
𝑦< 𝑥
3
c.
𝑦 ≤ 𝑥2 + 𝑥 − 6
2
𝑦< 𝑥
3
9-93. Complete the following inequalities that are graphed (fill each box with a number or symbol):
𝑦 ≤ 𝑥+3
𝑦 ≤ 2
𝑦 ≥
𝑦≥
𝑦 ≤
−2
𝑥−1
3
1
𝑥− 1
2
−2
𝑥+4
3
9-94. Match each graph below with the correct inequality.
a. y > −x + 2
b. y < 2x − 3
1
c. y > 2 x
2
d. y < − 3 x + 2
b.
d.
a.
c.
9-95. Solve each inequality below. Represent the solutions on number lines.
5𝑥 − 2 < 3
7x − 2 < 3 + 2x
a.
5𝑥 < 5
𝑥<1
x
−1
1
0
b.
2
3
1
c.
3(2m − 1) − 5m < −1 6𝑚 − 3 − 5𝑚 ≤ −1
m
0
1
d.
2
3
3≤1
2k + 3 < 2k + 1
x>2
3
𝑓𝑎𝑎𝑎𝑎
No Solution
k
x
5
𝑚≤2
4
𝑥≥6
4
𝑚 − 3 ≤ −1
6
7
−2
8
0
−1
1
2
9-96. Three years ago the average price of a movie ticket was $8.75 and now it is
$11.00. What was the annual multiplier and the percent increase?
11
𝑏 =
8.75
3
11
𝑏=�
≈ 1.079 ≈ 107.9%
8.75
3
The annual multiplier is
1.079 and the annual price
increase is 7.9%
9-97. Factor the following quadratics completely.
a.
b.
c.
5x3 + 13x2 − 6x = 𝑥(5𝑥 2 + 13𝑥 − 6) = 𝒙(𝟓𝟓 − 𝟐)(𝒙 + 𝟑)
6t2 − 26t + 8 = 2(3𝑡 2 − 13𝑡 + 4) = 𝟐(𝟑𝟑 − 𝟏)(𝒕 − 𝟒)
6x2 − 24 = 6(𝑥 2 − 4) = 𝟔(𝒙 + 𝟐)(𝒙 − 𝟐)
−2 −2𝑥
−6
5𝑥 5𝑥 2
15𝑥
−1 −1𝑡
+4
𝑥
3
3𝑡 3𝑡 2 −12𝑡
𝑡
−4
9-98. When a family with two adults and three children bought tickets for a
movie, they paid a total of $27.75. The next family in line, with two children
and three adults, paid $32.25 for the same movie. Find the adult and child
ticket prices by writing a system of equations and solving.
2𝑎 + 3𝑐 = 27.75
3𝑎 + 2𝑐 = 32.25
�
2 3 𝑎
27.75
�∙� �= �
�
3 2 𝑐
32.25
The adult ticket
costs $8.25 and
the child ticket
costs $3.75
9-99. Solve the equation below by completing the square.
Give the answer in exact (radical) form.
2
x2 − 6x + 3 = 0
𝑥 2 − 6𝑥 = −3
𝑥 − 6𝑥 + 9 = −3 + 9
(𝑥 − 3)2 = 6
−3 −3𝑥
𝑥
𝑥 − 3 = ±√6
𝑥2
𝑥
𝟗
−3𝑥
−3
𝒙 = 𝟑 ± √𝟔
9-100. Multiple Choice: Which of the points below is a solution of 𝑦 < |𝑥 − 3|?
1 < |2 − 3| 𝑖𝑖 𝑓𝑓𝑓𝑓𝑓
a.
(2, 1)
b.
c.
(−4, 5) 5 < |−4 − 3| 𝑖𝑖 𝑡𝑡𝑡𝑡
d.
(0, 3)
(−2, 8) 8 < |−2 − 3| 𝑖𝑖 𝑓𝑓𝑓𝑓𝑓
3 < |0 − 3| 𝑖𝑖 𝑓𝑓𝑓𝑓𝑓
PARCC PREP:
The positive root is
a. Label the positive root in the graph.
approximately
1.12 seconds,
What does the positive root reveal about
the amount of time it takes for
this problem situation?
Vertex
the ball to reach the ground
b. Label the vertex in the graph. What
does the vertex reveal about this
problem situation?
y-Intercept
Root
BONUS: Create an equation
for the graph in standard form
using the graphing calculator.
𝒚 = −𝟏𝟏𝒙𝟐 + 𝟏𝟏𝟏 + 𝟐
c.
Label the y-intercept in the graph. What
does the y-intercept reveal about this
problem situation?
d.
Use the graph to find the average rate
of change of the height of the ball from
0.25 seconds to 1 second.
The vertex is (0.5, 6).
It takes 0.5 seconds for the
ball to reach a maximum
height of 6 ft.
The y-intercept is 2.
The ball was hit at a
height of 2 feet.
Δ𝑦
Δ𝑥
−3
= 0.75 = −4
The average rate of
change of the height of
ball is -4 𝑓𝑓𝑓𝑓�𝑠𝑠𝑠.