Question

Sketch the graph of the inequality. \(y-4 x \leq 10\)

Short answer

The inequality \(y \leq 4x + 10\) in graph form is a solid upwards line beginning at the y-intercept (0,10), rising up to the right. The region below the line, inclusive of the line, is shaded to denote all solutions to the inequality.

Step-by-step solution

Rewrite the inequality in slope-intercept form

The first step involves rewriting the inequality in slope-intercept form (y=mx+b), where m represents the slope and b the y-intercept. Thus, the inequality \(y-4x\leq10\) can be rewritten as \(y \leq 4x + 10\). This represents a line with a slope of 4 and a y-intercept of 10.

Graph the line

After rewriting the inequality, graph the line using the slope and y-intercept. Begin at the y-intercept (0,10) and use the slope to find other points on the line. For this case, since the slope is 4, you rise 4 for every 1 unit you move to the right. Draw the line using these points. Since the inequality includes ‘less than or equal to’, the line will be solid, including the points on the line themselves in the region of solution.

Shade the Area

Finally, shade the area of the graph that satisfies the inequality. Since the inequality is \(y \leq 4x + 10\), this means that y is 'less than or equal to' 4x+10. Therefore, the region below the line (where y is less) should be shaded.