Q. Graph the inequality on the axes below. \(y \;\gt\; -x – 3\).

Q: What is the boundary line for the inequality y > -x - 3?

The boundary is the line y = -x - 3; graph it first, then use it to decide shading..

Q: Which side of the line do I shade?

Test a point (e.g., 0,0): 0 > -0 - 3 is true, so shade the region containing the origin (the half-plane above the line).

Q: What are the slope and y-intercept?

The slope is -1 and the y-intercept is (0,-3). From (0,-3) go right 1, down 1 to plot the line.

Q: How do I find the x- and y-intercepts?

Set y=0 to get 0=-x-3 so x-intercept (-3,0). Set x=0 to get y=-3 so y-intercept (0,-3).

Q: How can I check whether a point, say (2,-1) , satisfies the inequality?

How can I check whether a point, say (2,-1) , satisfies the inequality?

Q: How can I rewrite the inequality in standard/alternative forms?

Equivalent forms: x+y > -3 or x+y+3 > 0. These can help when comparing or combining inequalities.

Answer

Boundary line:
\[
y=-x-3
\]
Since the inequality is strict (y>−x−3), draw this line dashed. It passes through (0,−3) and (−3,0). Test point (0,0): 0>−3 is true, so shade the region above the line.

Final: dashed line through (0,−3) and (−3,0) with the half-plane above the line shaded.

Detailed Explanation

Graph the inequality y is greater than −x − 3 (y > −x − 3)

Write the boundary line. The boundary line is the equation you get by replacing the inequality sign with an equals sign:\[ y = -x – 3 \]
Put the line in slope-intercept form and read off slope and intercept. The equation is already in slope-intercept form \[ y = mx + b \] with\[ m = -1 \quad\text{(slope)}, \qquad b = -3 \quad\text{(y-intercept)}. \]So the line crosses the y-axis at the point (0, −3).
Plot two points and draw the boundary line (dashed). Start with the y-intercept point (0, −3). Use the slope m = −1 (rise / run = −1 / 1) to find another point:From (0, −3) move right 1 and down 1 to (1, −4). Alternatively, move left 1 and up 1 to (−1, −2).Because the inequality is strict (greater than, not greater than or equal to), draw this boundary line as a dashed line through these points.
Decide which side of the line to shade. Pick an easy test point not on the line, for example (0, 0). Substitute into the inequality \[ y > -x – 3 \]:\[ 0 > -0 – 3 \]\[ 0 > -3 \]This statement is true, so the region that contains (0,0) is the solution region. Therefore shade the half-plane on the same side of the dashed line as (0,0).
Final description of the graph. Draw the dashed line \[ y = -x – 3 \] through (0, −3) and (1, −4). Shade the region above that line (the half-plane containing (0,0)).In set notation the solution is \[ \{(x,y)\mid y > -x – 3\}. \]

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Algebra FAQs

What is the boundary line for the inequality \(y > -x - 3\)?

The boundary is the line \(y = -x - 3\); graph it first, then use it to decide shading..

Do I draw the boundary as solid or dashed?

Draw a dashed line because the inequality is strict (>), so points on \(y = -x - 3\) are not included..

Which side of the line do I shade?

Test a point (e.g., \(0,0\)): \(0 > -0 - 3\) is true, so shade the region containing the origin (the half-plane above the line).

What are the slope and y-intercept?

The slope is \(-1\) and the y-intercept is \((0,-3)\). From \((0,-3)\) go right 1, down 1 to plot the line.

How do I find the \(x\)- and \(y\)-intercepts?

Set \(y=0\) to get \(0=-x-3\) so x-intercept \((-3,0)\). Set \(x=0\) to get \(y=-3\) so y-intercept \((0,-3)\).

How can I check whether a point, say \( (2,-1) \), satisfies the inequality?

How can I rewrite the inequality in standard/alternative forms?

Equivalent forms: \(x+y > -3\) or \(x+y+3 > 0\). These can help when comparing or combining inequalities.

What changes if the inequality were \(y \ge -x - 3\)?

Use a solid boundary line (include points on the line) and shade the same side (test a point to confirm)..

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