Question

1. which of these expressions is equivalent to $3(x - 2)$?

a. $3x - 6$
c. $3x + 2$
b. $3x - 2$
d. $3x + 6$

which of these expressions is equivalent to $-2(x - 5)$?
a. $-2x - 5$
c. $-2x + 10$
b. $-2x + 5$
d. $-2x - 10$

for each expression, combine like terms, and write an equivalent expression with fewer terms.
a. $4x + 3x$
b. $3x + 5x - 1$
c. $5 + 2x + 7 + 4x$
d. $4 - 2x + 5x$
e. $10x - 5 + 3x - 2$

for each equation, find a value for $x$ that makes the equation true.
a. $x + 3 = 12$
d. $-4x = -24$
b. $2x + 3 = 20$
e. $2(x - 4) = 10$
c. $\frac{4}{3}x = \frac{10}{3}$
f. $-0.5x + 1.1 = -2$

Explanation:

First Question (Which of these expressions is equivalent to \( 3(x - 2) \)?)

Step 1: Apply the distributive property

The distributive property states that \( a(b - c) = ab - ac \). For \( 3(x - 2) \), we multiply 3 by \( x \) and 3 by \( -2 \).
\( 3(x - 2)=3\times x-3\times2 \)

Step 2: Simplify the multiplication

\( 3\times x = 3x \) and \( 3\times2 = 6 \), so the expression becomes \( 3x - 6 \).

Step 1: Apply the distributive property

Using the distributive property \( a(b - c)=ab - ac \), here \( a=-2 \), \( b = x \), \( c = 5 \). So \( -2(x - 5)=-2\times x-(-2)\times5 \)

Step 2: Simplify the multiplication

\( -2\times x=-2x \) and \( -(-2)\times5 = 10 \), so the expression becomes \( -2x + 10 \)

Step 1: Identify like terms

\( 4x \) and \( 3x \) are like terms (both have the variable \( x \) with power 1)

Step 2: Combine the coefficients

Add the coefficients of \( x \): \( 4 + 3=7 \), so \( 4x + 3x=7x \)

Answer:

A. \( 3x - 6 \)

Second Question (Which of these expressions is equivalent to \( -2(x - 5) \)?)