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Mathematics LibreTexts

3.4: Multiply and Divide Integers (Part 2)

  • Page ID
    6037
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

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    Evaluate Variable Expressions with Integers

    Now we can evaluate expressions that include multiplication and division with integers. Remember that to evaluate an expression, substitute the numbers in place of the variables, and then simplify.

    Example \(\PageIndex{10}\): evaluate

    Evaluate \(2x^2 − 3x + 8\) when \(x = −4\).

    Solution

    Substitute \(\textcolor{red}{-4}\) for x. \(2(\textcolor{red}{-4})^{2} - 3(\textcolor{red}{-4}) + 8\)
    Simplify exponents. \(2(16) - 3(-4) + 8\)
    Multiply. \(32 - (-12) + 8\)
    Subtract. \(44 + 8\)
    Add. \(52\)

    Keep in mind that when we substitute \(−4\) for \(x\), we use parentheses to show the multiplication. Without parentheses, it would look like \(2 • −4^2 − 3 • −4 + 8\).

    Exercise \(\PageIndex{19}\)

    Evaluate: \(3x^2 − 2x + 6\) when \(x = −3\)

    Answer

    \(39\)

    Exercise \(\PageIndex{20}\)

    Evaluate: \(4x^2-x-5\) when \(x = −2\)

    Answer

    \(13\)

    Example \(\PageIndex{11}\): evaluate

    Evaluate \(3x + 4y − 6\) when \(x = −1\) and \(y = 2\).

    Solution

    Substitute x = \(\textcolor{red}{-1}\) and y = \(\textcolor{blue}{2}\). \(3(\textcolor{red}{-1}) + 4(\textcolor{blue}{2}) - 6\)
    Multiply. \(-3 + 8 - 6\)
    Simplify. \(-1\)

    Exercise \(\PageIndex{21}\)

    Evaluate: \(7x + 6y − 12\) when \(x = −2\) and \(y = 3\)

    Answer

    \(-8\)

    Exercise \(\PageIndex{22}\)

    Evaluate: \(8x − 6y + 13\) when \(x = −3\) and \(y = −5\)

    Answer

    \(19\)

    Translate Word Phrases to Algebraic Expressions

    Once again, all our prior work translating words to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is product and for division is quotient.

    Example \(\PageIndex{12}\): translate

    Translate to an algebraic expression and simplify if possible: the product of \(−2\) and \(14\).

    Solution

    The word product tells us to multiply.

    Translate. (−2)(14)
    Simplify. −28

    Exercise \(\PageIndex{23}\)

    Translate to an algebraic expression and simplify if possible: the product of \(−5\) and \(12\)

    Answer

    \(-5(12)=-60\)

    Exercise \(\PageIndex{24}\)

    Translate to an algebraic expression and simplify if possible: the product of \(8\) and \(−13\)

    Answer

    \(8(-13)=-104\)

    Example \(\PageIndex{13}\)

    Translate to an algebraic expression and simplify if possible: the quotient of \(−56\) and \(−7\).

    Solution

    The word quotient tells us to divide.

    Translate. −56 ÷ (−7)
    Simplify. 8

    Exercise \(\PageIndex{25}\)

    Translate to an algebraic expression and simplify if possible: the quotient of \(−63\) and \(−9\)

    Answer

    \(-63 \div -9 = 7\)

    Exercise \(\PageIndex{26}\)

    Translate to an algebraic expression and simplify if possible: the quotient of \(−72\) and \(−9\)

    Answer

    \(-72 \div -9 = 8\)

    Key Concepts

    • Multiplication of Signed Numbers
      • To determine the sign of the product of two signed numbers:
        Same Signs Product
        Two positives
        Two negatives
        Positive
        Positive
        Different Signs Product
        Positive • negative
        Negative • positive
        Negative
        Negative
    • Division of Signed Numbers
      • To determine the sign of the quotient of two signed numbers:
        Same Signs Quotient
        Two positives
        Two negatives
        Positive
        Positive
        Different Signs Quotient
        Positive • negative
        Negative • Positive
        Negative
        Negative
    • Multiplication by \(-1\)
      • Multiplying a number by \(-1\) gives its opposite: \(-1a=-a\)
    • Division by \(-1\)
      • Dividing a number by \(-1\) gives its opposite: \(a \div (-1) = -a\)

    Practice Makes Perfect

    Multiply Integers

    In the following exercises, multiply each pair of integers.

    1. −4 • 8
    2. −3 • 9
    3. −5(7)
    4. −8(6)
    5. −18(−2)
    6. −10(−6)
    7. 9(−7)
    8. 13(−5)
    9. −1 • 6
    10. −1 • 3
    11. −1(−14)
    12. −1(−19)

    Divide Integers

    In the following exercises, divide.

    1. −24 ÷ 6
    2. −28 ÷ 7
    3. 56 ÷ (−7)
    4. 35 ÷ (−7)
    5. −52 ÷ (−4)
    6. −84 ÷ (−6)
    7. −180 ÷ 15
    8. −192 ÷ 12
    9. 49 ÷ (−1)
    10. 62 ÷ (−1)

    Simplify Expressions with Integers

    In the following exercises, simplify each expression.

    1. 5(−6) + 7(−2)−3
    2. 8(−4) + 5(−4)−6
    3. −8(−2)−3(−9)
    4. −7(−4)−5(−3)
    5. (−5)3
    6. (−4)3
    7. (−2)6
    8. (−3)5
    9. −42
    10. −62
    11. −3(−5)(6)
    12. −4(−6)(3)
    13. −4 • 2 • 11
    14. −5 • 3 • 10
    15. (8 − 11)(9 − 12)
    16. (6 − 11)(8 − 13)
    17. 26 − 3(2 − 7)
    18. 23 − 2(4 − 6)
    19. −10(−4) ÷ (−8)
    20. −8(−6) ÷ (−4)
    21. 65 ÷ (−5) + (−28) ÷ (−7)
    22. 52 ÷ (−4) + (−32) ÷ (−8)
    23. 9 − 2[3 − 8(−2)]
    24. 11 − 3[7 − 4(−2)]
    25. (−3)2−24 ÷ (8 − 2)
    26. (−4)2 − 32 ÷ (12 − 4)

    Evaluate Variable Expressions with Integers

    In the following exercises, evaluate each expression.

    1. −2x + 17 when (a) x = 8 (b) x = −8
    2. −5y + 14 when (a) y = 9 (b) y = −9
    3. 10 − 3m when (a) m = 5 (b) m = −5
    4. 18 − 4n when (a) n = 3 (b) n = −3
    5. p2 − 5p + 5 when p = −1
    6. q2 − 2q + 9 when q = −2
    7. 2w2 − 3w + 7 when w = −2
    8. 3u2 − 4u + 5 when u = −3
    9. 6x − 5y + 15 when x = 3 and y = −1
    10. 3p − 2q + 9 when p = 8 and q = −2
    11. 9a − 2b − 8 when a = −6 and b = −3
    12. 7m − 4n − 2 when m = −4 and n = −9

    Translate Word Phrases to Algebraic Expressions

    In the following exercises, translate to an algebraic expression and simplify if possible.

    1. The product of −3 and 15
    2. The product of −4 and 16
    3. The quotient of −60 and −20
    4. The quotient of −40 and −20
    5. The quotient of −6 and the sum of a and b
    6. The quotient of −7 and the sum of m and n
    7. The product of −10 and the difference of p and q
    8. The product of −13 and the difference of c and d

    Everyday Math

    1. Stock market Javier owns 300 shares of stock in one company. On Tuesday, the stock price dropped $12 per share. What was the total effect on Javier’s portfolio?
    2. Weight loss In the first week of a diet program, eight women lost an average of 3 pounds each. What was the total weight change for the eight women?

    Writing Exercises

    1. In your own words, state the rules for multiplying two integers.
    2. In your own words, state the rules for dividing two integers.
    3. Why is −24 ≠ (−2)4 ?
    4. Why is −42 ≠ (−4)2 ?

    Self Check

    (a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    (b) On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

    Contributors

    • Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (formerly of Santa Ana College). This content produced by OpenStax and is licensed under a Creative Commons Attribution License 4.0 license.