Final answer:
To graph the linear inequality 12x - 3y < 12, convert it to slope-intercept form y > mx + b. Then graph the boundary line and shade the region that satisfies the inequality.
Step-by-step explanation:
To graph the linear inequality 12x - 3y < 12, you need to first convert it to the slope-intercept form y > mx + b. In this case, divide both sides of the inequality by -3, resulting in -4x + y > -4. Next, graph the equation y = -4x - 4, which represents the boundary line of the inequality.
To graph the inequality, choose a test point not on the boundary line (for example, (0,0)), and substitute its coordinates into the original inequality. If the inequality is true, shade the region containing the test point. If the inequality is false, shade the region that does not contain the test point.
In this case, substituting (0,0) into the inequality results in -4(0) + 0 > -4, which simplifies to 0 > -4. Since 0 is not greater than -4, shade the region that does not contain the test point, which is the area below the boundary line.
The linear inequality 12x - 3y < 12 can be used to find all the ordered pairs that are solutions to it. To do this, first, we should rewrite the inequality in slope-intercept form, which is y = b + mx, where b represents the y-intercept and m represents the slope.
The given inequality 12x - 3y < 12 can be rearranged to -3y < -12x + 12 and then further to y > 4x - 4 after dividing all terms by -3 and reversing the inequality sign. When graphing this inequality, the line y = 4x - 4 will be dashed to indicate that points on the line are not included in the solution set, and the area above the line will be shaded because the inequality is greater than (>). The dashed line has a slope of 4 and a y-intercept of -4.