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Solve the first part of the inequality: 4 − 3x < x + 12.

Mathematics

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Answer

−2<x≤1-2 < x \le 1

94\frac{9}{4}

250 000 Tsh250\,000 \text{ Tsh}

Step-by-step solution

12 steps
  1. Solve the first part of the inequality

    4−3x<x+124 - 3x < x + 12.

    4−3x<x+124 - 3x < x + 12

    Subtract xx from both sides:

    4−4x<124 - 4x < 12

    Subtract 4 from both sides:

    −4x<8-4x < 8

    Divide by −4-4. Remember to reverse the inequality sign when dividing by a negative number:

    x>8−4x > \frac{8}{-4} x>−2x > -2
  2. Solve the second part of the inequality

    x+12≤−3x+292x + 12 \le \frac{-3x + 29}{2}.

    x+12≤−3x+292x + 12 \le \frac{-3x + 29}{2}

    Multiply both sides by 2:

    2(x+12)≤−3x+292(x + 12) \le -3x + 29 2x+24≤−3x+292x + 24 \le -3x + 29

    Add 3x3x to both sides:

    5x+24≤295x + 24 \le 29

    Subtract 24 from both sides:

    5x≤55x \le 5

    Divide by 5:

    x≤1x \le 1
  3. Combine the solutions.

    We have x>−2x > -2 and x≤1x \le 1. This can be written as:

    −2<x≤1-2 < x \le 1
  4. Illustrate the solution on a number line.

    The solution includes all numbers greater than −2-2 and less than or equal to 1. On a number line, this is represented by an open circle at −2-2 (since −2-2 is not included), a closed circle at 1 (since 1 is included), and a line segment connecting these two points.

    <-----|-----|-----|-----|-----|-----|-----|----->
         -3    -2    -1     0     1     2     3
               o------------------•
    

    The solution is −2<x≤1\boxed{-2 < x \le 1}.

  5. Evaluate each term in the expression.

    (127)13=(133)13=(3−3)13=3−3×13=3−1=13\left(\frac{1}{27}\right)^{\frac{1}{3}} = \left(\frac{1}{3^3}\right)^{\frac{1}{3}} = (3^{-3})^{\frac{1}{3}} = 3^{-3 \times \frac{1}{3}} = 3^{-1} = \frac{1}{3} 256×9=256×9=16×3=48\sqrt{256 \times 9} = \sqrt{256} \times \sqrt{9} = 16 \times 3 = 48 729−1=1729729^{-1} = \frac{1}{729} 722=(8×9)2=82×92=64×8172^2 = (8 \times 9)^2 = 8^2 \times 9^2 = 64 \times 81
  6. Substitute the evaluated terms back into the expression.

    13×481729×722\frac{\frac{1}{3} \times 48}{\frac{1}{729} \times 72^2}
  7. Simplify the numerator.

    13×48=16\frac{1}{3} \times 48 = 16
  8. Simplify the denominator.

    1729×722=193×(8×9)2=193×82×92=829=649\frac{1}{729} \times 72^2 = \frac{1}{9^3} \times (8 \times 9)^2 = \frac{1}{9^3} \times 8^2 \times 9^2 = \frac{8^2}{9} = \frac{64}{9}
  9. Divide the simplified numerator by the simplified denominator.

    16649=16×964\frac{16}{\frac{64}{9}} = 16 \times \frac{9}{64} =16×964= \frac{16 \times 9}{64} =1×94= \frac{1 \times 9}{4} =94= \frac{9}{4}

    The evaluated expression is 94\boxed{\frac{9}{4}}.

  10. Calculate the amount in Kenya Shillings (Ksh) the tourist received from changing South African Rands (SAR).

    On 12/1/2024, the tourist sells SAR to the bank, so the bank buys SAR.
    Buying rate for 1 SAR = 7.08 Ksh.
    Amount in Ksh = 15 000 SAR×7.08 Ksh/SAR=106 200 Ksh15\,000 \text{ SAR} \times 7.08 \text{ Ksh/SAR} = 106\,200 \text{ Ksh}.

  11. Calculate the remaining Kenya Shillings after spending.

    Amount spent = 53 075 Ksh.
    Remaining Ksh = 106 200 Ksh−53 075 Ksh=53 125 Ksh106\,200 \text{ Ksh} - 53\,075 \text{ Ksh} = 53\,125 \text{ Ksh}.

  12. Calculate the amount in Tanzanian Shillings (Tsh) the tourist received from changing the remaining Kenya Shillings.

    On 12/2/2024, the tourist buys Tsh from the bank, so the bank sells Tsh.
    Selling rate for 100 Tanzanian Shillings = 21.25 Ksh.
    This means 1 Tsh = 21.25100 Ksh=0.2125 Ksh\frac{21.25}{100} \text{ Ksh} = 0.2125 \text{ Ksh}.
    Amount in Tsh = Remaining KshKsh per Tsh\frac{\text{Remaining Ksh}}{\text{Ksh per Tsh}}
    Amount in Tsh = 53 125 Ksh0.2125 Ksh/Tsh=250 000 Tsh\frac{53\,125 \text{ Ksh}}{0.2125 \text{ Ksh/Tsh}} = 250\,000 \text{ Tsh}.

    The amount received in Tanzanian shillings is 250 000 Tsh\boxed{250\,000 \text{ Tsh}}.

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