Question

problem solving

1. an nhl hockey player has 59 goals so far in a season. what are the possible numbers of additional goals the player can score to match or break the nhl record of 92 goals in a season?

reasoning

1. which two of the following inequalities are equivalent to the inequality $x - b < 3$, where $b$ is a constant? justify your answer.

a $x - b - 3 < 0$
b $0 > b - x + 3$
c $x < 3 - b$
d $-3 < b - x$

mathematical connections

1. write and solve an inequality to find the possible values of $x$.

perimeter $\leq 18.7$
4.1 ft
4.9 ft
$x$ ft
6.4 ft

Explanation:

Problem 29

Step1: Define variable for extra goals

Let $x$ = number of additional goals.

Step2: Set up inequality for record

Total goals need $\geq 92$, so $59 + x \geq 92$.

Step3: Solve for $x$

$x \geq 92 - 59$
$x \geq 33$

Step1: Analyze Option A

Subtract 3 from both sides of $x-b<3$:
$x-b-3 < 3-3$
$x-b-3 < 0$, which matches A.

Step2: Analyze Option B

Rewrite original inequality: $x-b<3 \implies -x+b > -3 \implies b-x > -3 \implies b-x+3 > 0$, which contradicts B ($0 > b-x+3$).

Step3: Analyze Option C

Solve original inequality for $x$: $x < 3 + b$, which does not match C ($x < 3 - b$).

Step4: Analyze Option D

Rewrite original inequality: $x-b<3 \implies -x+b > -3 \implies b-x > -3$, which is equivalent to $-3 < b-x$, matching D.

Step1: Write perimeter inequality

Sum all sides, set $\leq 18.7$:
$4.9 + 4.1 + x + 6.4 \leq 18.7$

Step2: Simplify left side

$15.4 + x \leq 18.7$

Step3: Solve for $x$

$x \leq 18.7 - 15.4$
$x \leq 3.3$

Answer:

The player needs to score 33 or more additional goals, so $x \geq 33$ where $x$ is a non-negative integer.

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Problem 31