The system of linear inequalities is represented by the graph are [tex]\boxed{y > x - 2}[/tex] and [tex]\boxed{y < x + 1}.[/tex] Option (a) is correct.
Further explanation:
The linear equation with slope m and intercept c is given as follows.
[tex]\boxed{y = mx + c}[/tex]
The formula for slope of line with points [tex]\left( {{x_1},{y_1}} \right)[/tex] and [tex]\left( {{x_2},{y_2}}\right)[/tex] can be expressed as,
[tex]\boxed{m = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}}[/tex]
Given:
The given options are as follows,
(a). [tex]y > x - 2{\text{ and }}y < x + 1[/tex]
(b). [tex]y < x - 2{\text{ and }}y > x + 1[/tex]
(c).[tex]y < x - 2{\text{ and }}y > x + 1[/tex]
(d). [tex]y > x - 2{\text{ and }}y < x + 1[/tex]
Explanation:
The first line passes the points [tex]\left( {0, 1}\right)[/tex] and [tex]\left( {-1,0}\right).[/tex]
The slope of the line can be obtained as follows.
[tex]\begin{aligned}m&=\frac{{0 - 1}}{{0 - 1}}\\&=\frac{{ - 1}}{{ - 1}}\\ &= 1\\\end{gathered}[/tex]
The slope of the line is [tex]m = 1.[/tex]
The line intersect y-axis at [tex]\left( {1,1}\right)[/tex]. Therefore, y-intercept is 1.
Substitute [tex]\left( {0,0}\right)[/tex] in equation [tex]y< x+1[/tex] to check whether the equation includes origin.
[tex]\begin{aligned}0&<0 + 1\\0&< 1\\\end{aligned}[/tex]
The equation of first line is [tex]y < x + 1.[/tex]
The second line passes the points [tex]\left( {0, -2}\right)[/tex] and [tex]\left({2,0}\right).[/tex]
The slope of the line can be obtained as follows.
[tex]\begin{aligned}m&=\frac{{0 - \left({ - 2} \right)}}{{2 - 0}}\\&=\frac{2}{2}\\&= 1\\\end{aligned}[/tex]
The slope of the line is [tex]m = 1.[/tex]
The line intersect y-axis at [tex]\left( {-2,0} \right).[/tex] Therefore, y-intercept is -2.
Substitute [tex]\left( {0,0}\right)[/tex] in equation [tex]y > x - 2[/tex] to check whether the equation includes origin.
[tex]\begin{aligned}0&> 0- 2\\0&> - 2\\\end{aligned}[/tex]
The equation of second line is [tex]y > x -2.[/tex]
The system of linear inequalities is represented by the graph are [tex]\boxed{y > x - 2}[/tex] and [tex]\boxed{y < x + 1}.[/tex]
Option (a) is correct.
Option (b) is not correct.
Option (c) is not correct.
Option (d) is not correct.
Learn more:
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2. Learn more about equation of circle brainly.com/question/1506955.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Linear equation
Keywords: numbers, slope, slope intercept, inequality, equation, linear inequality, shaded region, y-intercept, graph, representation, origin.
