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A team scores n goals and the other team scores n+5 goals. The total number of goals is less than 15. What is the possible range of n?

Mathematics

ScanSolve AI Answer

Answer

n<5n < 5

128∘128^\circ

Step-by-step solution

8 steps
  1. Formulate the total number of goals.

    Team A scores nn goals.
    Team B scores (n+5)(n+5) goals.
    The total number of goals scored by both teams is the sum of their individual scores:

    Total goals=n+(n+5)\text{Total goals} = n + (n+5)
  2. Formulate the inequality.

    The total number of goals is less than 15.

    n+(n+5)<15n + (n+5) < 15
  3. Solve the inequality for nn.

    Combine like terms:

    2n+5<152n + 5 < 15

    Subtract 5 from both sides:

    2n<15−52n < 15 - 5 2n<102n < 10

    Divide by 2:

    n<102n < \frac{10}{2} n<5n < 5

    The inequality is n<5\boxed{n < 5}.

  4. Identify the properties of a parallelogram.

    In a parallelogram RSTU, opposite angles are equal, and consecutive angles are supplementary (add up to 180∘180^\circ).
    Given:
    ∠RUT=(2x+30)∘\angle RUT = (2x+30)^\circ
    ∠RST=(7x−25)∘\angle RST = (7x-25)^\circ

  5. Use the property of opposite angles to find xx.

    Angles ∠RUT\angle RUT (at vertex U) and ∠RST\angle RST (at vertex S) are opposite angles in the parallelogram RSTU. Therefore, they are equal.

    ∠RUT=∠RST\angle RUT = \angle RST 2x+30=7x−252x+30 = 7x-25
  6. Solve for xx.

    Subtract 2x2x from both sides:

    30=7x−2x−2530 = 7x - 2x - 25 30=5x−2530 = 5x - 25

    Add 25 to both sides:

    30+25=5x30 + 25 = 5x 55=5x55 = 5x

    Divide by 5:

    x=555x = \frac{55}{5} x=11x = 11
  7. Calculate the measure of ∠RST\angle RST.

    Substitute x=11x=11 into the expression for ∠RST\angle RST:

    ∠RST=(7x−25)∘=(7(11)−25)∘\angle RST = (7x-25)^\circ = (7(11)-25)^\circ ∠RST=(77−25)∘\angle RST = (77-25)^\circ ∠RST=52∘\angle RST = 52^\circ
  8. Calculate the value of ∠STU\angle STU.

    Angles ∠STU\angle STU (at vertex T) and ∠RST\angle RST (at vertex S) are consecutive angles in the parallelogram RSTU. Therefore, they are supplementary.

    ∠STU+∠RST=180∘\angle STU + \angle RST = 180^\circ

    Substitute the value of ∠RST\angle RST:

    ∠STU+52∘=180∘\angle STU + 52^\circ = 180^\circ

    Subtract 52∘52^\circ from both sides:

    ∠STU=180∘−52∘\angle STU = 180^\circ - 52^\circ ∠STU=128∘\angle STU = 128^\circ

    The value of angle STU is 128∘\boxed{128^\circ}.

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