Podcast
Questions and Answers
Which statements are true for the expressions 5 + 3x - 2 and x + 2(x + 1) + 1 when x = 3 and x = 7? (Check all that apply)
Which statements are true for the expressions 5 + 3x - 2 and x + 2(x + 1) + 1 when x = 3 and x = 7? (Check all that apply)
- The expressions are never equivalent.
- The expressions have equivalent values for any value of x. (correct)
- When x = 10, both expressions have a value of 33. (correct)
- The expressions have equivalent values when x + 6 (correct)
Which statements are true for the expressions 7x + 4 and 3x + 5 + 4x - 1 when x = 4 and x = 6? (Check all that apply)
Which statements are true for the expressions 7x + 4 and 3x + 5 + 4x - 1 when x = 4 and x = 6? (Check all that apply)
- The expressions have equivalent values for any value of x. (correct)
- The expressions have equivalent values if x = 8. (correct)
- The expressions are never equivalent.
- When x = 2, both expressions have a value of 18.
Which expression is equivalent to 7(xy)?
Which expression is equivalent to 7(xy)?
x(7y)
Which expressions are equivalent to 8.9x + 6.2 + 8.7? (Check all that apply)
Which expressions are equivalent to 8.9x + 6.2 + 8.7? (Check all that apply)
Which expressions are equivalent when m = 1 and m = 4?
Which expressions are equivalent when m = 1 and m = 4?
What proves that the expressions 4(x + 3) + 2 and 6(x + 2) are equivalent?
What proves that the expressions 4(x + 3) + 2 and 6(x + 2) are equivalent?
What describes the correct method to determine if 4x - x + 5 and 8 - 3x - 3 are equivalent?
What describes the correct method to determine if 4x - x + 5 and 8 - 3x - 3 are equivalent?
Which expressions are equivalent to g + h + (j + k)? (Check all that apply)
Which expressions are equivalent to g + h + (j + k)? (Check all that apply)
Did Vance apply the associative property correctly in (4.5m + 7/8) - 9 as 4.5m(7/8 - 9)?
Did Vance apply the associative property correctly in (4.5m + 7/8) - 9 as 4.5m(7/8 - 9)?
What property was applied to create the equivalent expression 7x + (5 + 6x) -----> (7x + 5) + 6x?
What property was applied to create the equivalent expression 7x + (5 + 6x) -----> (7x + 5) + 6x?
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Study Notes
Equivalent Expressions
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Equivalent expressions have the same value for any specific value of the variable involved, such as x or m.
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For x=3 and x=7:
- Expressions 5 + 3x - 2 and x + 2(x + 1) + 1 are equivalent since both yield the same result, specifically for x + 6.
- Both expressions evaluate to 33 when x=10.
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For x=4 and x=6:
- Expressions 7x + 4 and 3x + 5 + 4x - 1 are found to be equivalent.
- Their values equal 18 when x=2, and they maintain equivalence at x=8.
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The expression 7(xy) can be equivalent to x(7y) due to the commutative property of multiplication.
Expressions Simplification
- The expression 8.9x + 6.2 + 8.7 can be simplified equivalently to:
- 8.9x + 8 - 7 + 6.2
- 8.7 + 8.9x + 6.2
- 6.2 + 8.7 + 8.9x
Determining Equivalence
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For expressions 3m + 4 and m + 4 + 2m:
- They are equivalent for m=1 and m=4.
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To prove equivalence of 4(x + 3) + 2 and 6(x + 2):
- Evaluation at x=1 yields 18, and at x=8 the result is 60, confirming equivalence.
Evaluation Method
- To determine if two expressions (e.g., 4x - x + 5 and 8 - 3x - 3) are equivalent:
- Use substitution and evaluate each expression with two different values.
- The expressions are equivalent if they yield the same results for both values substituted.
Properties of Expressions
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The expression 7x + (5 + 6x) can be rewritten as (7x + 5) + 6x.
- This transformation demonstrates the use of the associative property, which allows grouping without changing the sum.
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Incorrect application of the associative property can occur when crucial operators (like addition) are omitted in simplifications, as seen in Vance's transformation of (4.5m + 7/8) - 9 to 4.5m(7/8 - 9).
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