- Which property lets you remove parentheses when a factor is outside them? - The outside factor applies to each term inside the parentheses. - After removing the parentheses, check that each original term has been accounted for.

Distribute \(5\) to both terms inside the parentheses: \(5(x + 2) = 5x + 10\).

\(5x + 10\)

- Which two terms must the outside factor multiply? - What happens to each sign when you multiply by a negative number?

1. Multiply both terms in the parentheses by \(-2\). 2. The result is \(-6x+8\).

\(-6x+8\)

- Which terms contain the variable? - Which terms are numbers without a variable?

1. Combine the terms with \(x\): \(3x+2x=5x\). 2. Combine the constants: \(5-8=-3\). The expression becomes \(5x-3\).

\(5x-3\)

- How could you rewrite the expression with parentheses? - What must be true for expressions to be equivalent for every input?

1. Distribute in the first expression: \(4(x+1)=4x+4\). 2. Both expressions have the same value for every \(x\), so they are equivalent.

Yes. \(4(x+1)=4x+4\) for every \(x\) by the distributive property.

- Remove the parentheses in \(f\) using the distributive property. - Combine constant terms separately from variable terms. - Arrange each result with the \(x\)-term first and the constant second.

1. Distribute and combine constants in \(f\): \(f(x)=2x-6+4=2x-2\). 2. Combine constants in \(g\): \(g(x)=-\frac{1}{2}x+5-2=-\frac{1}{2}x+3\).

\(f(x)=2x-2\) \(g(x)=-\frac{1}{2}x+3\)

- Write the rectangle's area as length times width. - Use the distributive property to remove the parentheses.

1. The area is \(4(x + 3)\). 2. Distribute \(4\) to both terms inside the parentheses: \(4x + 12\).

\(4x + 12\) square units

- Check whether the outside factor affects every term inside the parentheses. - Compare the original and rewritten expressions term by term.

1. The rewrite is not correct because the factor \(5\) must multiply both terms inside the parentheses. 2. Distributing gives \(5(2x) - 5(3) = 10x - 15\).

No. The correct equivalent expression is \(10x - 15\). Cian multiplied only the first term by \(5\).

- Remove the parentheses before combining terms. - Combine variable terms with variable terms and constants with constants.

1. Distribute \(3\): \(6x + 15 - 4x + 7\). 2. Combine like terms: \(6x - 4x = 2x\) and \(15 + 7 = 22\). 3. The simplified expression is \(2x + 22\).

\(2x + 22\)

- What does equivalent mean beyond a single test value? - Can you choose an easy input different from \(2\)? - What would one pair of unequal results show?

1. When \(x=0\), the first expression equals \(6\) and the second equals \(4\). 2. Since the expressions differ at \(x=0\), they are not equivalent for every \(x\). One matching input is not enough.

No. At \(x=0\), the values are \(6\) and \(4\). They are not equivalent.

- Rewrite the expression with parentheses first. - What must stay the same for every value of \(x\)? - Compare the simplified expression with the form containing \(a\).

1. Rewrite the first expression: \(2(x+5)+3x=2x+10+3x=5x+10\). 2. Its coefficient of \(x\) must equal \(a\). Thus \(a=5\), and both expressions are \(5x+10\) for every \(x\).

\(a=5\). Both expressions simplify to \(5x+10\).

- What does it mean for two expressions to be equal for every value of \(x\)? - Rewrite the expression with \(k\) so its variable term and constant are easy to see. - Would checking just one value of \(x\) prove the two expressions equivalent?

1. Rewrite the first expression: \(k(x+4)+2x=(k+2)x+4k\). 2. Equivalent linear expressions have the same coefficient of \(x\) and the same constant term. 3. Comparing coefficients gives \(k+2=7\), so \(k=5\). Comparing constants gives \(4k=20\), also yielding \(k=5\). 4. With \(k=5\), both expressions simplify to \(7x+20\) for every \(x\).

\(k=5\). The coefficients match because \(k+2=7\), and the constants match because \(4k=20\).

- A rectangle has two lengths and two widths. - Write the perimeter expression before simplifying it. - Distribute first, then combine like terms.

1. Use the perimeter formula: \(P = 2(2x + 1) + 2(x - 2)\). 2. Distribute: \(P = 4x + 2 + 2x - 4\). 3. Combine like terms: \(P = 6x - 2\).

\(6x - 2\) units