Question

question 8 (1 point) (01.03 mc)
the perimeter of a rectangle can be found using the equation ( p = 2l + 2w ), where ( p ) is the perimeter, ( l ) is the length, and ( w ) is the width of the rectangle. can the perimeter of the rectangle be 60 units when its width is 12 units and its length is 18 units?
○ a no. if the rectangle has ( l = 18 ) and ( w = 12 ), ( p ) would not equal 60.
○ b no. the rectangle cannot have ( p = 60 ) and ( l = 18 ) because ( l + w ) is less than 24.
○ c yes. the rectangle can have ( p = 60 ) and ( l = 18 ) because ( 60 = 24 + 18 ).
○ d yes. the rectangle can have ( p = 60 ) and ( l = 18 ) because ( p = 2(18) + 2(12) ) would equal 60.

question 9 (1 point) (01.04 lc)
choose the missing step in the given solution to the inequality ( -x - 3 < 13 + 3x )
( -x - 3 < 13 + 3x )
( -4x < 16 )
( x > -4 )
options:
○ a ( -4x - 3 > 13 )
○ b ( 2x - 3 < 13 )
○ c ( -4x - 3 < 13 )
○ d ( 2x - 3 > 13 )

Explanation:

Question 8

To determine if the perimeter can be 60, use the formula \( P = 2L + 2W \). Substitute \( L = 18 \) and \( W = 12 \): \( P = 2(18) + 2(12) = 36 + 24 = 60 \). So the rectangle can have \( P = 60 \) with \( L = 18 \) and \( W = 12 \).

Step 1: Start with the inequality

We have the inequality \(-x - 3 < 13 + 3x\). First, we want to get all the \(x\) terms on one side. Let's add \(x\) to both sides.
\(-x - 3 + x < 13 + 3x + x\)
Simplifying the left side: \(-3 < 13 + 4x\)

Step 2: Isolate the term with \(x\)

Now, subtract 13 from both sides.
\(-3 - 13 < 13 + 4x - 13\)
Simplifying both sides: \(-16 < 4x\)

Step 3: Solve for \(x\)

Divide both sides by 4.
\(\frac{-16}{4} < \frac{4x}{4}\)
Simplifying: \(-4 < x\) or \(x > -4\)
But we need to find the missing step between \(-x - 3 < 13 + 3x\) and \(x > -4\). Let's go back. After \(-x - 3 < 13 + 3x\), add \(x\) to both sides: \(-3 < 13 + 4x\), then subtract 13: \(-16 < 4x\) (which is \(-4x < 16\) if we rearrange, but let's check the options). Wait, maybe another approach. Let's start from \(-x - 3 < 13 + 3x\), add \(x\) to both sides: \(-3 < 13 + 4x\), then subtract 13: \(-16 < 4x\) (or \(-4x < 16\) by moving \(4x\) to left and \(-3\) to right: \(-x - 3 - 3x < 13\) → \(-4x - 3 < 13\), then add 3: \(-4x < 16\), which is one of the steps. Wait, the given steps are: first \(-x - 3 < 13 + 3x\), then the missing step, then \(-4x < 16\), then \(x > -4\). So let's find the step between \(-x - 3 < 13 + 3x\) and \(-4x < 16\). Subtract \(3x\) and add \(x\) to both sides? Wait, subtract \(3x\) from both sides: \(-x - 3 - 3x < 13\) → \(-4x - 3 < 13\), then add 3: \(-4x < 16\). Wait, no, the options are: a. \(-4x - 3 > 13\), b. \(2x - 3 < 13\), c. \(-4x - 3 < 13\), d. \(2x - 3 > 13\). Wait, let's do the algebra correctly. Starting with \(-x - 3 < 13 + 3x\). Let's move all \(x\) terms to the left and constants to the right. Subtract \(3x\) from both sides: \(-x - 3 - 3x < 13\) → \(-4x - 3 < 13\). So that's option c? Wait no, option c is \(-4x - 3 < 13\)? Wait the options: a. \(-4x - 3 > 13\), b. \(2x - 3 < 13\), c. \(-4x - 3 < 13\), d. \(2x - 3 > 13\). Wait, when we subtract \(3x\) from both sides of \(-x - 3 < 13 + 3x\), we get \(-x - 3 - 3x < 13\), which simplifies to \(-4x - 3 < 13\), which is option c? Wait no, let's check again. Wait the original inequality is \(-x - 3 < 13 + 3x\). Let's add \(x\) to both sides: \(-3 < 13 + 4x\). Then subtract 13: \(-16 < 4x\) (or \(-4x < 16\) by multiplying both sides by -1, but that reverses the inequality). Wait, maybe I made a mistake. Let's look at the steps given: first step \(-x - 3 < 13 + 3x\), then missing step, then \(-4x < 16\), then \(x > -4\). So to get from \(-x - 3 < 13 + 3x\) to \(-4x < 16\), we can subtract \(3x\) and add \(x\) to both sides? Wait, subtract \(3x\) from both sides: \(-x - 3 - 3x < 13\) → \(-4x - 3 < 13\), then add 3 to both sides: \(-4x < 16\). Ah, so the missing step is \(-4x - 3 < 13\), which is option c? Wait no, option c is \(-4x - 3 < 13\)? Wait the options: c. \(-4x - 3 < 13\)? Wait the user's options: a. \(-4x - 3 > 13\), b. \(2x - 3 < 13\), c. \(-4x - 3 < 13\), d. \(2x - 3 > 13\). Wait, when we subtract \(3x\) from both sides of \(-x - 3 < 13 + 3x\), we get \(-x - 3 - 3x < 13\) → \(-4x - 3 < 13\), which is option c. Then from \(-4x - 3 < 13\), add 3: \(-4x < 16\), then divide by -4 (reversing inequality): \(x > -4\). So the missing step is \(-4x - 3 < 13\), which is option c.

Step 1: Analyze the inequality transformation

Given the inequality \(-x - 3 < 13 + 3x\), we need to find the intermediate step before \(-4x < 16\).
Subtract \(3x\) from both sides:
\(-x - 3 - 3x < 13 + 3x - 3x\)
Simplify: \(-4x - 3 < 13\) (this matches option c).

Answer:

d. Yes. The rectangle can have \( P = 60 \) and \( L = 18 \) because \( P = 2(18) + 2(12) \) would equal 60.

Question 9