Question

Sketch the graph of the inequality. $$-2 x \geq 10$$

Short answer

The graph of the inequality \( -2x \geq 10 \) is a vertical line at \( x = -5 \), with all the area to the left of this line shaded to indicate the set of solutions.

Step-by-step solution

Simplify the Inequality for x

First, the inequality needs to be simplified to find the value for x. This involves dividing both sides of the inequality by -2 (which is the coefficient of x). It should be noted that when we divide or multiply an inequality by a negative number, the direction of the inequality sign changes. Thus, the inequality -2x >= 10 simplifies to x <= -5.

Plot the line x = -5

The inequality \(x \leq -5\) represents all values of x that are less than or equal to -5. This can be represented graphically by a vertical line which passes through the point x = -5 on the x-axis. As equal to -5 is included (since it's \(\leq -5\) and not \(< -5\)), the line is drawn as a filled line, not dotted.

Indicate the solution to the inequality on the graph

As the inequality \( x \leq -5 \) represents all values of \( x \) that are less than or equal to -5, it means all points on the left-hand side of the line \( x = -5 \) are included in the solution. This can be indicated on the graph by shading the area to the left of the line \( x = -5 \).