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Mathematics LibreTexts

2.6: Exercise 2

  • Page ID
    13972
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    State the property or indicate that no property is applicable.

    1. \(5+x=x+5\)

    2. \(7(a-2)=7a-7(2)\)

    3. \(5(x\cdot 2)=(5x)(2)\)

    4. \((y+2)+4=4+(y+2)\)

    5. \(6\cdot (t\cdot 2)=(6\cdot t)\cdot (6\cdot 2)\)

    6. \(w-5=5-w\)

    7. \(xy=yx\)

    8. \(a\div \pi=\pi \div a\)

    9. \((-4.7)+[(6.5)+(\sqrt{2})]=[(-4.7)+(6.5)]+(\sqrt{2})\)

    10. \(x-4=4-x\)

    11. Factor \(12x+18\)

    12. Factor \(24x^2+12x\)

    The solutions follow:

    Keep them covered up till you have worked out each of the problems above.

    1. \(5+x=x+5\)
      Solution:
      \(5+x=x+5\) illustrates the commutative property of addition. The terms commuted to each other’s position.

    2. \(7(a-2)=7a-7(2)\)
      Solution:
      \(7(a-2)=7a-7(2)\) illustrates the distributive property of multiplication over addition.

    3. \(5(x\cdot 2)=(5x)(2)\)
      Solution:
      \(5(x\cdot 2)=(5x)(2)\) illustrates the associative property of multiplication.

    4. \((y+2)+4=4+(y+2)\)
      Solution:
      \((y+2)+4=4+(y+2)\) illustrates the commutative property of addition. \((y+2)\) and 4 have commuted to each other’s position. \(y\) remains grouped with 2.

      N regrouping took place.

    5. \(6\cdot (t\cdot 2)=(6\cdot t)\cdot (6\cdot 2)\)
      Solution:
      \(6\cdot (t\cdot 2)=(6t)\cdot (6\cdot 2)\) does not illustrate any of the mentioned properties. You cannot distribute multiplication over multiplication.

    6. \(w-5=5-w\)
      Solution:
      \(w-5=5-w\) does not illustrate any of the mentioned properties. Subtraction is not commutative.
      We shall learn later that \(w-5=w+(-5)=(-5)+w\) because addition is commutative.

    7. \(xy=yx\)
      Solution:
      \(xy=yx\) illustrates the commutative property of multiplication. The terms commuted to each other’s position.

    8. \(a\div \pi=\pi \div a\)
      Solution:
      \(a\div \pi=\pi \div a\) does not illustrate any of the mentioned properties. Division is not commutative.

    9. \((-4.7)+[(6.5)+(\sqrt{2})]=[(-4.7)+(6.5)]+(\sqrt{2})\)
      Solution:
      \((-4.7)+[(6.5)+(\sqrt{2})]=[(-4.7)+(6.5)]+(\sqrt{2})\) illustrates the associative property of addition. \(6.5\) is associated (grouped) with \(\sqrt{2}\) on the left and \(-4.7\) on the right.

    10. \(x-4=4-x\)
      Solution:
      \(x-4=4-x\) does not illustrate any of the mentioned properties. Subtraction is not commutative.

    11. Factor \(12x+18\)
      Solution:
      \(\begin{array}{rcl lll} 12x+18&=&(3)(4)x+(3)(6)\\ &=&3(4x+6)\\ \end{array}\)

      A quick check: \(3(4x+6)=3(4x)+3(6)=12x+18\)Gthe result grees wiht the iginal statement.

    12. Factor \(24x^2+12x\)
      Solution:
      \(\begin{array}{rcl lll} 24x^2+12x&=&2(12)x\ x+1(12)x\\ &=&12x(2x+1)\\ \end{array}\)
      Note the \(1\) besides the \(12\).

      A quick check: \(12x(2x+1)=12x(2x)+12x(1)=24x^2+12x\)