P.11. Linear Inequalities
Thus far, we have only discussed algebraic equations. What about inequalities? The simplest type of equalities are linear inequalities in one variable. (Linear inequalities in two or more variables are considerably more complicated and require graphical solutions in two or more dimensions.) The general forms of a linear inequality in one variable are ax + b > c, ax + b ≥ c, ax + b ≤ c, or ax + b < c, where a, b, and c are arbitrary real constants with a nonzero.
The solutions to linear inequalities can be expressed graphically using the number line or using interval notation. To represent an interval graphically, we draw a solid line above the number line corresponding to the points lying within the interval. We also draw an open or closed circle at the endpoint(s) of the interval, open if the point is contained in the interval and closed otherwise. Interval notation expresses an interval in any of the four forms (a,b), [a,b), (a,b], or [a,b], where a and b are the endpoints of the interval with a<b. A parentheis indicates that the corresponding point does not belong to the interval and a bracket indicates that it does. If the interval goes indefinitely to the left, the left endpoint is -∞ and if it goes indefinitely to the right, the right enpoint is ∞. The endpoints ±∞ are always proceeded or followed by a parenthesis, indicating that these points are not included in the interval.
Example 1: Solve the linear inequality 3x + 4 ≥ 10.
Solution: Subtracting 4 from both sides, we obtain 3x ≥ 6. Dividing both sides by 3, we obtain x ≥ 2. The interval notation for this solution set is [2,∞). The solution set is a ray with endpoint at x = 2 and pointing to the right as shown:
Example 2: Solve the linear inequality 5 - 2x < 11.
Solution: Subtracting 5 from both sides, we obtain -2x < 6. Next we divide both sides by -2, noting that the direction of the inequality must change since we are multiplying or dividing by a negative number. We obtain x > -3. The interval notation for this solution set is (-3,∞).The solution set is an open ray with (missing) endpoint at x = -3 and pointing to the right as shown:
Often one is presented with two simultaneous linear inequalities, such as the following example
Example 3: Solve the linear inequalities 3x + 8 ≥ 5 and 10 - x > 7.
Solution: We solve these inequalities one at a time. The first inequality simplifies to x ≥ -1 and the second to x < 3. The interval notation for this solution set is [-1,3).The solution set is thus the intersection of the rays corresponding to each of these inequalities, which is a line segment with endpoint x = -1 and missing endpoint at x = 3 as shown: