Answer:
- y ≤ x +1
- y ≤ -5/4x +5
- y ≥ -2
Explanation:
You want the inequalities that define the shaded region in the given graph.
Lines
The graph shows 3 lines, one each with positive, negative, and zero slope.
There are several ways we could write the equations for these lines. We can use the slope-intercept form, as that is probably the most familiar.
Slope-intercept form
The slope-intercept form of the equation of a line is ...
y = mx + b . . . . . . . where m is the slope, and b is the y-intercept.
Slope
The slope is the ratio of "rise" to "run" for the line. We can find these values by counting the grid squares vertically and horizontally between points where the line crosses grid intersections.
The line with positive slope (up to the right) crosses the x-axis at -1 and the y-axis at +1. It has 1 unit of rise and 1 unit of run between those points. Its slope is ...
m = 1/1 = 1
The line with negative slope (down to the right) crosses the x-axis at x=4 and the y-axis at y=5. It has -5 units of rise for 4 units of run between those points. Its slope is ...
m = -5/4
The horizontal line has no rise, so its slope is 0. It is constant at y = -2.
Intercept
As we have already noted, the line with positive slope intersects the y-axis at +1. Its equation will be ...
y = x +1
The line with negative slope intersects the y-axis at +5. Its equation will be ...
y = -5/4x +5
The line with zero slope has a y-intercept of -2, so its equation is ...
y = -2. . . . . . . . . . mx = 0x = 0
Shading
The boundary lines are all drawn as solid lines, so the inequality will include the "or equal to" case for all of them.
When shading is below the line, the form of the inequality is y ≤ ( ).
When shading is above the line, the form of the inequality is y ≥ ( ).
Shading is below the two lines with non-zero slope, and above the line with zero slope.
The inequalities are ...
- y ≤ x +1
- y ≤ -5/4x +5
- y ≥ -2
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Additional comment
The intercept form of the equation for a line is ...
x/a +y/b = 1 . . . . . . . . . . 'a' = x-intercept; 'b' = y-intercept
Using the intercepts we identified above, the three boundary line equations could be ...
- x/-1 +y/1 = 1 . . . . . . line with positive slope
- x/4 +y/5 = 1 . . . . . . line with negative slope
- y/-2 = 1 . . . . . . . . . . line with 0 slope; has no x-intercept
These can be turned to inequalities by considering the shading in either the vertical direction (above/below), or the horizontal direction (left/right).
When the coefficient of y is positive, and the shading is above, the inequality will look like ... y ≥ .... If shading is to the right, and the coefficient of x is positive, the inequality will look like ... x ≥ .... If the shading is reversed or the coefficient is negative (but not both), the direction of the inequality will change.
Considering this, we could write the three inequalities as ...
x/-1 +y/1 ≤ 1; x/4 +y/5 ≤ 1; y/-2 ≤ 1
These could be rearranged to a more pleasing form, but the point here is to give you another way to look at the problem.
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