Question

Check whether \((0,0)\) is a solution. Then sketch the graph of the inequality. $$ x+y \geq-1 $$

Short answer

Indeed, the point (0,0) is a solution to the inequality \(x+y \geq -1\). The graph of the inequality is a solid line passing through the points (-1,0) and (0,-1), with the region on and above the line shaded.

Step-by-step solution

Checking the Solution

Replace \(x\) and \(y\) in the inequality \(x+y \geq -1\) with the co-ordinates of the given point \((0,0)\). If that makes the inequality true, then the point is a solution.

Graphing and Shading the Inequality

Draw a two-dimensional graph. Because the inequality is \(\geq\), the graph will be a solid line. Start by plotting the line \(x + y = -1\). Then choose a random point that is not on the line, say \((0,0)\) because it is usually the simplest. Replace \(x\) and \(y\) in the inequality with these new co-ordinates. If true, shade the side of the line that contains \((0,0)\), otherwise shade the other side.

Interpreting the Graph

The shaded region of the graph represents the solution to the inequality \(x + y \geq -1\). Any point in this region will satisfy the inequality.

Key concepts

Linear Inequality Solutions

When dealing with linear inequalities such as \( x + y \geq -1 \), finding solutions is similar to solving equations, but with a focus on a range of values rather than a single solution. Here, the inequality includes every point \((x,y)\) in the Cartesian plane that makes the statement \(x+y\geq-1\) true.

To test a specific point, like \((0,0)\), you substitute it into the inequality: \(0+0\geq-1\) simplifies to \(0\geq-1\), which is true. Hence, \((0,0)\) is a solution. Importantly, because this is an inequality, it has many solutions forming a region in the plane, rather than a single point on a graph like you'd find with an equation.

Inequality Graphing

Graphing inequalities differs from graphing equations since it involves shading a region that represents all the possible solutions. To graph the inequality \(x+y\geq-1\), you start by plotting its corresponding line, \(x+y=-1\). This line divides the plane into two halves.

As the inequality symbol is \(\geq\), you will draw a solid line to indicate that points on the line are part of the solution set. If the inequality were strict, as in \(x+y>-1\), the line would be dashed, signaling that points exactly on the line aren't included in the solution set. With linear inequalities, always remember that the graph is as important as the algebra involved in solving them.

Shading Inequalities

Shading is the key visual cue distinguishing a solution area for an inequality. Once you've graphed the line for \(x+y=-1\), you need to establish which side of the line to shade.

To decide on the shading, choose a test point not on your line (\((0,0)\) is a great choice for simplicity) and substitute it into the original inequality. If the inequality holds true with the test point, then the region containing that point is shaded. In our problem, since \((0,0)\) made \(x+y\geq-1\) true, we shade the side of the line where \((0,0)\) is located. The shading visually expresses all points that satisfy the inequality.

Testing Solutions for Inequalities

The final verification of understanding an inequality involves testing its solutions. After graphing and shading, as in steps 1 and 2, you want to ensure your shaded region is correct.

Select random points from both the shaded and non-shaded regions and substitute them into the original inequality. If a point from the shaded area makes the inequality true and one from the non-shaded area does not, your shading is correct. This step is a way to check your work and further grasp the significance of the inequality's solution set on the graph. Always test points to solidify your understanding of the solutions to inequalities.