Question

simplify the expression below completely.
$10 - 4(2x + 7)$
...
ve the perimeter of the rectangle below in
simplest form.
rectangle image with length $3x - 7$ and width $x + 2$
the equation below.
$5x + 1 = -49$

Explanation:

First Sub - Question: Simplify \(10 - 4(2x + 7)\)

Step 1: Distribute the - 4

We use the distributive property \(a(b + c)=ab+ac\). Here \(a=-4\), \(b = 2x\) and \(c = 7\). So we have \(10-4\times(2x)-4\times(7)\) which simplifies to \(10-8x - 28\).

Step 2: Combine like terms

Combine the constant terms \(10\) and \(-28\). \(10-28=-18\). So the simplified expression is \(-8x-18\).

Step 1: Recall the perimeter formula for a rectangle

The perimeter \(P\) of a rectangle is given by \(P = 2\times(\text{length}+\text{width})\). Substituting the given length \(l=3x - 7\) and width \(w=x + 2\) into the formula, we get \(P=2[(3x - 7)+(x + 2)]\).

Step 2: Simplify the expression inside the brackets

Combine like terms inside the brackets: \((3x - 7)+(x + 2)=3x+x-7 + 2=4x-5\).

Step 3: Distribute the 2

Multiply \(2\) with \((4x - 5)\) using the distributive property: \(2\times(4x)-2\times(5)=8x-10\).

Step 1: Subtract 1 from both sides

To isolate the term with \(x\), we subtract \(1\) from both sides of the equation. \(5x+1 - 1=-49 - 1\), which simplifies to \(5x=-50\).

Step 2: Divide both sides by 5

Divide both sides of the equation \(5x=-50\) by \(5\). \(\frac{5x}{5}=\frac{-50}{5}\), so \(x = - 10\).

Answer:

\(-8x - 18\)

Second Sub - Question: Find the perimeter of the rectangle with length \(3x - 7\) and width \(x + 2\)