Answer:
see attached
Explanation:
The boundary lines and solution region are shown on the attached graph. It is convenient to let a graphing calculator plot them.
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If you'd like to plot the boundary lines by hand, it is useful to note the equations for them are in standard form:
ax +by = c
This means the slope is m = -a/b, and the y-intercept is c/b. Here, the odd constant values mean that (most) intercepts will involve fractions.
2x+5y<19
slope: -2/5
y-intercept: 19/5 = 3.8
points on the line: (7, 1), (2, 3)
The general form ...
x < ( ) . . . . means the line is dashed and shading is to the left.
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x-2y<-9
slope: 1/2
y-intercept: 9/2 = 4.5
points on the line: (1, 5), (-3, 3)
The general form ...
x < ( ) . . . . means the line is dashed and shading is to the left.
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Solution Region
The lines have slopes of opposite signs, so cross in a fashion that forms an "X" on the graph. Shading is left of both lines, so will be in the left quadrant of the X. The boundary lines are not part of the solution region.
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Additional comments
The boundary line will be solid when the inequalitiy includes the "or equal to" case. This is identified by the symbols ≤ or ≥. Here, we have the symbol <, which does not include the "or equal to" case, so the boundary line is dashed. (The same would be true for an inequality using >.)
To determine shading direction, it is convenient to pick a variable with a positive coefficient, and look at its relation to the inequality symbol. In the first inequality, we could choose either of ...
- x < ( ) . . . . . shading left of the dashed line
- y < ( ) . . . . . shading below the dashed line
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In the second inequality, the only variable with a positive coefficient is x, and its relation to the inequality symbol is ...
- x < ( ) . . . . . shading left of the dashed line
If we were to rewrite the second inequality to give the y-variable a positive coefficient, we could get either of ...
- y > ( ) . . . . . multiply by -1/2
- ( ) < y . . . . . add the opposite of the y-term
Either way, the shading is above the dashed line.
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About points on the line:
The inequalities have odd constants (19, -9) and coefficients with different parity (one even, one odd). That makes it convenient to choose an odd value for the variable with the odd coefficient. The result will be an integer value for the variable with the even coefficient.