Question

1. the sum of three consecutive odd integers is $-105$. liam, jonas, and nora each define a variable and write an equation for the situation.

liam
let $x$ represent the smallest integer.
$x+(x+1)+(x+2)=-105$

jonas
let $x$ represent the smallest integer.
$x+(x+1)+(x+3)=-105$

nora
let $x$ represent the smallest integer.
$x+(x+2)+(x+4)=-105$

a. whose equation is correct? explain.

b. use the correct equation to find the integers.

1. for three consecutive odd integers, the largest integer is nine more than twice the smallest integer. what are the integers?

Explanation:

For Problem 2a:

Consecutive odd integers differ by 2. If \(x\) is the smallest odd integer, the next is \(x+2\), and the largest is \(x+4\). Liam's equation uses consecutive integers (difference 1), Jonas uses inconsistent differences, so Nora's equation correctly models consecutive odd integers.

Step1: Simplify the correct equation

Combine like terms in \(x+(x+2)+(x+4)=-105\):
\(3x + 6 = -105\)

Step2: Isolate the variable term

Subtract 6 from both sides:
\(3x = -105 - 6\)
\(3x = -111\)

Step3: Solve for \(x\)

Divide both sides by 3:
\(x = \frac{-111}{3} = -37\)

Step4: Find the other integers

Calculate the second integer: \(x+2 = -37 + 2 = -35\)
Calculate the third integer: \(x+4 = -37 + 4 = -33\)

Step1: Define variables

Let \(x\) = smallest odd integer, \(x+4\) = largest odd integer.

Step2: Set up the equation

Translate the problem to math:
\(x+4 = 2x + 9\)

Step3: Solve for \(x\)

Subtract \(x\) from both sides:
\(4 = x + 9\)
Subtract 9 from both sides:
\(x = 4 - 9 = -5\)

Step4: Find the other integers

Second integer: \(x+2 = -5 + 2 = -3\)
Largest integer: \(x+4 = -5 + 4 = -1\)

Answer:

Nora's equation is correct. Consecutive odd integers have a difference of 2, so the three integers are \(x\), \(x+2\), and \(x+4\), which matches her equation.

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For Problem 2b: