A two-step inequality gets its name from the two steps necessary to solve it, usually subtraction of a constant followed by division of the coefficient. Single-variable inequalities can be demonstrated graphically on a number line using shaded arrows to indicate the values the variable can have and dots to demarcate the boundaries. A single inequality is marked by a dot and an arrow; a system of inequalities will have two dots and one or two arrows.
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Look at the side of your inequality that has the unknown variable. If there is a constant term being added to (or subtracted from) the variable term, subtract (or add) the constant to both sides of the inequality. This will cancel out the constant on the side with the variable. For example, given -5x - 3 < 12, you would add 3 to both sides, leaving -5x < 15 without changing the inequality.
Look at the variable term. If it has a coefficient (number that the variable is multiplied by), divide both sides of the equation by that value to cancel out the coefficient. For example, you would divide both sides of -5x < 15 by -5, leaving x < 5.
Reverse the direction of the inequality only if you divided by a negative number in Step 2. Otherwise, proceed to Step 4. Since we divided by -5 in Step 2, we would change the inequality to x > 5.
Draw a number line with zero somewhere near the middle, and the value on the right side of the inequality in Step 3 marked by a hatch mark at the corresponding position on the number line (left is negative values, right is positive values). Fill in other values with hatch marks if you desire.
Draw a circle on the hatch mark on the line corresponding to the value from Step 3. If the inequality is "less than" or "greater than" (< or >), leave the circle unfilled. If the inequality is "less than or equal to" or "greater than or equal to" (≤ or ≥), fill in the circle entirely.
Draw an arrow emanating along the number line--to the right of the dot if it is a "greater than" inequality, to the left of the dot if it is a "less than" inequality. Write your variable name above the arrow.
Perform Steps 1 through 3 from above to solve for the unknown variable in both inequalities.
Draw a number line with zero, the term on the right from your first in equality, and the term on the right from your second inequality all marked with hatch marks.
Draw circles on the two values on the number line from your inequalities. If the inequality is "less than" or "greater than" (< or >), leave the circle unfilled. If the inequality is "less than or equal to" or "greater than or equal to" (≤ or ≥), fill in the circle entirely.
Draw arrows emanating along the number line--to the right of the dot if it is a "greater than" inequality, to the left of the dot if it is a "less than" inequality. If the two arrows meet in the number line, do not draw arrow heads, and instead shade in the entire segment, with the variable name written above. If the arrows head in opposite directions, draw arrow heads on both ends and write the variable name above both arrows.
Tips and warnings
- You can only graph inequalities with a single unknown on a number line. Inequalities with two unknowns must be graphed on a Cartesian grid with x and y axes.
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