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1. Sketch the set of points

(x, y)

which satisfy the following relations. a) ...

Feb 8, 2024

1. Sketch the set of points

(x, y)

which satisfy the following relations. a)

0 \leq y \leq 2 x

and

0 \leq x \leq 2

\quad y / 2 \leq x \leq 2

and

0 \leq y \leq 4

Solution by Steps

step 1

To sketch the set of points for the first relation, we need to understand the inequalities. The first inequality

0 \leq y \leq 2x

represents the area between the line

y = 2x

and the x-axis. The second inequality

0 \leq x \leq 2

represents the vertical boundaries from

x = 0

x = 2

step 2

Plot the line

y = 2x

and shade the region below it, since

y

is less than or equal to

2x

. Then, draw vertical lines at

x = 0

and

x = 2

and shade the region between them. The intersection of these shaded regions is the solution set for the first relation

step 3

To sketch the set of points for the second relation, we need to understand the inequalities. The first inequality

y/2 \leq x \leq 2

represents the area between the line

y = 2x

(reflected over the line

y = x

) and the vertical line

x = 2

. The second inequality

0 \leq y \leq 4

represents the horizontal boundaries from

y = 0

y = 4

step 4

Plot the line

y = 2x

and reflect it over the line

y = x

to get

x = y/2

. Then, draw the vertical line at

x = 2

and shade the region between this line and the curve

x = y/2

. Finally, draw horizontal lines at

y = 0

and

y = 4

and shade the region between them. The intersection of these shaded regions is the solution set for the second relation

Answer

The sketches for the set of points satisfying the given relations are completed as per the steps above.

Key Concept

Graphical representation of inequalities

Explanation

To sketch the set of points that satisfy a system of inequalities, we plot the lines or curves corresponding to each inequality and shade the appropriate regions. The solution set is the intersection of these shaded regions.