Question

1. select the graph of the system of inequalities.

$x - y \leq 1$
$x + 2y < 4$
a
graph a
c
graph c
b
graph b
d
graph d

1. part a

a theater charges $x$ for adult tickets and $y$ for child tickets. four adult tickets and 3 child tickets cost $156. three adult tickets and 4 child tickets cost $145. write and solve a system of equations to find the adult and child ticket prices.
adult: $\square$ child: $\square$
part b
a family of 1 adult and 2 children has a budget of $60. do they have enough money to go to the theater?
$\square$ yes
$\square$ no

1. a shop sells hardcover copies of a book for $22 and paperback copies for $15. yesterday, the shop had sales of the book of no more than $350. which inequality shows the number of copies that were sold?

a $22x + 15y > 350$
b $22x + 15y < 350$
c $22x + 15y \geq 350$
d $22x + 15y \leq 350$

Explanation:

Question 11

Step1: Analyze \( x - y \leq 1 \)

Rewrite as \( y \geq x - 1 \). This is a solid line (since \( \leq \)) with slope 1, y-intercept -1. The region is above the line.

Step2: Analyze \( x + 2y < 4 \)

Rewrite as \( y < -\frac{1}{2}x + 2 \). This is a dashed line (since \( < \)) with slope \( -\frac{1}{2} \), y-intercept 2. The region is below the line.

Step3: Match with graphs

Check which graph has a solid line \( y = x - 1 \) (above it) and dashed line \( y = -\frac{1}{2}x + 2 \) (below it). Graph C matches.

Step1: Set up equations

From "4 adult, 3 child cost $156": \( 4x + 3y = 156 \)
From "3 adult, 4 child cost $145": \( 3x + 4y = 145 \)

Step2: Solve the system

Multiply first equation by 4: \( 16x + 12y = 624 \)
Multiply second equation by 3: \( 9x + 12y = 435 \)
Subtract: \( 7x = 189 \Rightarrow x = 27 \)
Substitute \( x = 27 \) into \( 4x + 3y = 156 \): \( 108 + 3y = 156 \Rightarrow 3y = 48 \Rightarrow y = 16 \)

Step1: Calculate cost for 1 adult, 2 children

Cost = \( x + 2y = 27 + 2(16) = 27 + 32 = 59 \)

Step2: Compare with budget

Budget is $60. \( 59 \leq 60 \), so they have enough.

Answer:

C

Question 12 - Part A