Problems from Linear inequations of two variables

An wine-producing industry produces wine and vinegar. Twice the production of wine is always smaller than or equal to than the vinegar production plus four units. Determine the region of validity of the inequation (and to draw it).

a) Identify the variables.

b) Express the restriction as an inequation of the variables.

c) Give the expression of the straight line associated with the restriction (and draw it).

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Development:

a) $x =$ production of wine. $y =$ production of vinegar.

b) $2 \cdot x ⩽ y + 4 \Rightarrow 2 \cdot x - y ⩽ 4$

c) $2 \cdot x = y + 4 \Rightarrow y = 2 x - 4$

Trying point $(x = 0, y = 0)$ it is seen that the inequation is satisfied: $2 \cdot 0 - 0 ⩽ 4$ . Therefore the validity region is the semiplane over the straight line.

Solution:

The validity region is the semiplane over the straight line.

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A company makes two hat models: Bae and Viz. The manufacture of every Bae model needs $2$ hours to manufacture, while that of the model Viz needs $3$ hours. The manufacturing section of molded has a $1500$ hour a month maximum. Determine the region of validity of the inequation (and to draw it).

a) Identify the variables.

b) Express the restriction as an inequation of the variables.

c) Give the expression of the straight line associated with the restriction (and draw it).

See development and solution

Development:

a) $x =$ number of hats Bae. $y =$ number of hats Viz.

b) $2 \cdot x + 3 \cdot y ⩽ 1500$

c) $2 \cdot x + 3 \cdot y = 1500 \Rightarrow 3 \cdot y = - 2 \cdot x + 1500 \Rightarrow y = - \frac{2}{3} \cdot x + 500$

Trying point $(x = 0, y = 0)$ eit is seen that the inequation is satisfied: $2 \cdot 0 + 3 \cdot 0 ⩽ 1500$ . Therefore the validity region is the semiplane below the straight line.

Solution:

The validity region is the semiplane below the straight line.

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