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solving inequalities in algebra

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Notice that there are two inequalities with an "AND" between the two inequalities.  (This is called a "compound inequality".  A compound inequality contains two or more inequalities that are separated by either "AND" or "OR".)

What does this mean?  It means that after you have solved for x in the two inequalities, we are going to take the intersection of the solutions.

Also, solving an inequality is similar to solving equations except that you have to switch the inequality symbols (<, >, <=, >=) when multiplying or dividing by negative numbers.

What does "solving an inequality" (or "solving an equation", for that matter) mean?  (Here, we are talking about inequalities (or equations) with ONE variable.)  It means isolating the variable on one side of the inequality (or equation) and "moving" everything else to the other side.


So let's look at the first inequality: 5x - 3 < 7.  (We want x on one side and everything else on the other side.)

The first step is to add 3 to both sides of the inequality: 5x  - 3 + 3 < 7 + 3.  (Homework: Why isn't the first step dividing by 5?)

We get 5x < 10.

The last step is to divide both sides by 5: 5x/5 < 10/5.  (Note that the inequality, <, does not change since we are dividing by a positive number.)

We get x < 2 as the solution to the first inequality.


Now let's look at the second inequality: 2 - x < x + 6.  (Here, there are x's on both sides so we need to combine the two x's at some point.)

There are at least four possible first steps here: one could (1) subtract 2 from both sides, (2) add x to both sides, (3) subtract x from both sides, or (4) subtract 6 from both sides.

Here, I shall pick choice (2) and add x to both sides: 2 - x + x < x + 6 + x.  (This combines the two x's on the right-hand side of the inequality and ensures that the coefficient of x is positive so there is no need to switch inequality symbols.)

We get 2 < 2x + 6.

Then the next step is to subtract 6 from both sides: 2 - 6 < 2x + 6 - 6.

We get -4 < 2x.

The last step is to divide both sides by 2: -4/2 < 2x/2.

We get -2 < x as the solution to the second inequality.


The final solution is the intersection of x < 2 (the set of all real numbers less than 2) AND -2 < x (this second inequality can also be written as x > -2) (the set of all real numbers greater than -2): the set of all real numbers between -2 (excluding -2) and 2 (excluding 2).  This can be written as -2 < x < 2.


One can check the answer by choosing a number between -2 and 2, let's say 0, and plugging it into both of the original inequalities and checking to see that the inequalities are correct.  Let's look at the first inequality: 5(0) - 3 < 7.  This simplifies to -3 < 7, which is a true statement.  Now, let's look at the second inequality: 2 - 0 < 0 + 6.  This simplifies to 2 < 6, which is also a true statement.  (Homework: plug in a number outside of the interval, -2 < x < 2, say -3 (or 3), and show that one of the resulting inequalities is a false statement.)

5x-3 < 7

5x < 7+3

5x < 10

x < 2


2-x < x+6

-x - x < 6-2

-2x < 4

multiplying both sides of the equation by -1 gives:

2x < -4

x < -2