Question

linear equations, inequalities, and systems section f checkpoint
1 draw a graph of a system of inequalities that has no solution.
standard
a - rei.d.12:
2 to have a valid password on a website, the password must have at least 2 letters, at least 1 number, and be less than 8 characters (letters or numbers) in length.
a. write 3 inequalities to represent the constraints. make sure to include what each variable represents.
b. would the password j8675309 be a solution to the inequalities you wrote? explain your reasoning.
c. graph the solution region for the system of 3 inequalities.
standard
a - ced.a.3:
standard
a - rei.d.12:
algebra 1 cc by nc 2024
illustrative mathematics® 1

Explanation:

Problem 1

Step1: Pick parallel lines with gaps

Choose $y > x + 2$ and $y < x - 1$.

Step2: Graph boundary lines

First, draw dashed lines $y = x + 2$ and $y = x - 1$ (dashed because inequalities are strict).

Step3: Shade solution regions

Shade above $y = x + 2$ and below $y = x - 1$.

Step1: Define variables

Let $L$ = number of letters, $N$ = number of numbers, $T$ = total characters ($T = L + N$).

Step2: Write inequality for letters

At least 2 letters: $L \geq 2$

Step3: Write inequality for numbers

At least 1 number: $N \geq 1$

Step4: Write inequality for total length

Less than 8 characters: $T = L + N < 8$

Step1: Count password components

Password j8675309: 1 letter ($L=1$), 7 numbers ($N=7$), total length $T=8$.

Step2: Check inequality 1

$L \geq 2$? $1 \geq 2$ is false.

Step3: Check inequality 3

$L + N < 8$? $8 < 8$ is false.

Answer:

A system like $\boldsymbol{y > x + 2}$ and $\boldsymbol{y < x - 1}$ has no overlapping solution region, so no solution. The graph has two parallel dashed lines with non-overlapping shaded areas.

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Problem 2a