Solved on Dec 13, 2023

Find the ordered pair $(x,y)$ that satisfies the constraints: $x + y \geq 4$ and $2x + y \leq 7$ , where $x$ represents hot dogs at $\$2$ each and $y$ represents peanuts at $\$1$ each, given Cody has $\$7$ .

STEP 1

Assumptions
1. Cody has $7.<br />2. He wants to buy at least 4 snacks.<br />3. Hot dogs (x) cost$ 2 each.
4. Peanuts (y) cost $1 each.<br />5. The inequalities$ x + y \geq 4 $and$ 2x + y \leq 7$ must be satisfied by the ordered pair.

STEP 2

We need to check if each ordered pair satisfies both inequalities. Let's start with the ordered pair $(3,1)$ .

STEP 3

Substitute $x=3$ and $y=1$ into the first inequality $x + y \geq 4$ .
$3 + 1 \geq 4$

STEP 4

Check if the inequality holds true.
$4 \geq 4$

STEP 5

Since $4$ is equal to $4$ , the first inequality is satisfied by the ordered pair $(3,1)$ .

STEP 6

Now, substitute $x=3$ and $y=1$ into the second inequality $2x + y \leq 7$ .
$2(3) + 1 \leq 7$

STEP 7

Calculate the left side of the inequality.
$6 + 1 \leq 7$

STEP 8

Check if the inequality holds true.
$7 \leq 7$

STEP 9

Since $7$ is equal to $7$ , the second inequality is also satisfied by the ordered pair $(3,1)$ .

STEP 10

Now let's check the ordered pair $(5,4)$ . Substitute $x=5$ and $y=4$ into the first inequality $x + y \geq 4$ .
$5 + 4 \geq 4$

STEP 11

Check if the inequality holds true.
$9 \geq 4$

STEP 12

Since $9$ is greater than $4$ , the first inequality is satisfied by the ordered pair $(5,4)$ .

STEP 13

Now, substitute $x=5$ and $y=4$ into the second inequality $2x + y \leq 7$ .
$2(5) + 4 \leq 7$

STEP 14

Calculate the left side of the inequality.
$10 + 4 \leq 7$

STEP 15

Check if the inequality holds true.
$14 \leq 7$

STEP 16

Since $14$ is not less than or equal to $7$ , the second inequality is not satisfied by the ordered pair $(5,4)$ .

STEP 17

Based on the checks, the ordered pair $(3,1)$ satisfies both inequalities, while the ordered pair $(5,4)$ does not.
The solution is the ordered pair $(3,1)$ .

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Find the ordered pair $(x,y)$ that satisfies the constraints: $x + y \geq 4$ and $2x + y \leq 7$ , where $x$ represents hot dogs at $\$2$ each and $y$ represents peanuts at $\$1$ each, given Cody has $\$7$ .

STEP 1

Assumptions
1. Cody has $7.<br />2. He wants to buy at least 4 snacks.<br />3. Hot dogs (x) cost$ 2 each.
4. Peanuts (y) cost $1 each.<br />5. The inequalities$ x + y \geq 4 $and$ 2x + y \leq 7$ must be satisfied by the ordered pair.

STEP 2

We need to check if each ordered pair satisfies both inequalities. Let's start with the ordered pair $(3,1)$ .

STEP 3

Substitute $x=3$ and $y=1$ into the first inequality $x + y \geq 4$ .
$3 + 1 \geq 4$

STEP 4

Check if the inequality holds true.
$4 \geq 4$

STEP 5

Since $4$ is equal to $4$ , the first inequality is satisfied by the ordered pair $(3,1)$ .

STEP 6

Now, substitute $x=3$ and $y=1$ into the second inequality $2x + y \leq 7$ .
$2(3) + 1 \leq 7$

STEP 7

Calculate the left side of the inequality.
$6 + 1 \leq 7$

STEP 8

Check if the inequality holds true.
$7 \leq 7$

STEP 9

Since $7$ is equal to $7$ , the second inequality is also satisfied by the ordered pair $(3,1)$ .

STEP 10

Now let's check the ordered pair $(5,4)$ . Substitute $x=5$ and $y=4$ into the first inequality $x + y \geq 4$ .
$5 + 4 \geq 4$

STEP 11

Check if the inequality holds true.
$9 \geq 4$

STEP 12

Since $9$ is greater than $4$ , the first inequality is satisfied by the ordered pair $(5,4)$ .

STEP 13

Now, substitute $x=5$ and $y=4$ into the second inequality $2x + y \leq 7$ .
$2(5) + 4 \leq 7$

STEP 14

Calculate the left side of the inequality.
$10 + 4 \leq 7$

STEP 15

Check if the inequality holds true.
$14 \leq 7$

STEP 16

Since $14$ is not less than or equal to $7$ , the second inequality is not satisfied by the ordered pair $(5,4)$ .

STEP 17

Based on the checks, the ordered pair $(3,1)$ satisfies both inequalities, while the ordered pair $(5,4)$ does not.
The solution is the ordered pair $(3,1)$ .

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Find the ordered pair (x,y)(x,y)(x,y) that satisfies the constraints: x+y≥4x + y \geq 4x+y≥4 and 2x+y≤72x + y \leq 72x+y≤7, where xxx represents hot dogs at $2\$2$2 each and yyy represents peanuts at $1\$1$1 each, given Cody has $7\$7$7.

STEP 1

STEP 2

We need to check if each ordered pair satisfies both inequalities. Let's start with the ordered pair (3,1)(3,1)(3,1).

STEP 3

Substitute x=3x=3x=3 and y=1y=1y=1 into the first inequality x+y≥4x + y \geq 4x+y≥4.3+1≥43 + 1 \geq 43+1≥4

STEP 4

Check if the inequality holds true.4≥44 \geq 44≥4

STEP 5

Since 444 is equal to 444, the first inequality is satisfied by the ordered pair (3,1)(3,1)(3,1).

STEP 6

Now, substitute x=3x=3x=3 and y=1y=1y=1 into the second inequality 2x+y≤72x + y \leq 72x+y≤7.2(3)+1≤72(3) + 1 \leq 72(3)+1≤7

STEP 7

Calculate the left side of the inequality.6+1≤76 + 1 \leq 76+1≤7

STEP 8

Check if the inequality holds true.7≤77 \leq 77≤7

STEP 9

Since 777 is equal to 777, the second inequality is also satisfied by the ordered pair (3,1)(3,1)(3,1).

STEP 10

Now let's check the ordered pair (5,4)(5,4)(5,4). Substitute x=5x=5x=5 and y=4y=4y=4 into the first inequality x+y≥4x + y \geq 4x+y≥4.5+4≥45 + 4 \geq 45+4≥4

STEP 11

Check if the inequality holds true.9≥49 \geq 49≥4

STEP 12

Since 999 is greater than 444, the first inequality is satisfied by the ordered pair (5,4)(5,4)(5,4).

STEP 13

Now, substitute x=5x=5x=5 and y=4y=4y=4 into the second inequality 2x+y≤72x + y \leq 72x+y≤7.2(5)+4≤72(5) + 4 \leq 72(5)+4≤7

STEP 14

Calculate the left side of the inequality.10+4≤710 + 4 \leq 710+4≤7

STEP 15

Check if the inequality holds true.14≤714 \leq 714≤7

STEP 16

Since 141414 is not less than or equal to 777, the second inequality is not satisfied by the ordered pair (5,4)(5,4)(5,4).

STEP 17

Based on the checks, the ordered pair (3,1)(3,1)(3,1) satisfies both inequalities, while the ordered pair (5,4)(5,4)(5,4) does not.The solution is the ordered pair (3,1)(3,1)(3,1).

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Find the ordered pair $(x,y)$ that satisfies the constraints: $x + y \geq 4$ and $2x + y \leq 7$ , where $x$ represents hot dogs at $\$2$ each and $y$ represents peanuts at $\$1$ each, given Cody has $\$7$ .

We need to check if each ordered pair satisfies both inequalities. Let's start with the ordered pair $(3,1)$ .

Substitute $x=3$ and $y=1$ into the first inequality $x + y \geq 4$ .
$3 + 1 \geq 4$

Check if the inequality holds true.
$4 \geq 4$

Since $4$ is equal to $4$ , the first inequality is satisfied by the ordered pair $(3,1)$ .

Now, substitute $x=3$ and $y=1$ into the second inequality $2x + y \leq 7$ .
$2(3) + 1 \leq 7$

Calculate the left side of the inequality.
$6 + 1 \leq 7$

Check if the inequality holds true.
$7 \leq 7$

Since $7$ is equal to $7$ , the second inequality is also satisfied by the ordered pair $(3,1)$ .

Now let's check the ordered pair $(5,4)$ . Substitute $x=5$ and $y=4$ into the first inequality $x + y \geq 4$ .
$5 + 4 \geq 4$

Check if the inequality holds true.
$9 \geq 4$

Since $9$ is greater than $4$ , the first inequality is satisfied by the ordered pair $(5,4)$ .

Now, substitute $x=5$ and $y=4$ into the second inequality $2x + y \leq 7$ .
$2(5) + 4 \leq 7$

Calculate the left side of the inequality.
$10 + 4 \leq 7$

Check if the inequality holds true.
$14 \leq 7$

Since $14$ is not less than or equal to $7$ , the second inequality is not satisfied by the ordered pair $(5,4)$ .

Based on the checks, the ordered pair $(3,1)$ satisfies both inequalities, while the ordered pair $(5,4)$ does not.
The solution is the ordered pair $(3,1)$ .