Solve Compound Inequalities with “and”

Now that we know how to solve linear inequalities, the next step is to look at compound inequalities. A compound inequality is made up of two inequalities connected by the word “and” or the word “or.” For example, the following are compound inequalities.

[begin{array} {lll} {x+3>−4} &{text{and}} &{4x−5leq 3} {2(y+1)<0} &{text{or}} &{y−5geq −2} end{array} nonumber]

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To solve a compound inequality means to find all values of the variable that make the compound inequality a true statement. We solve compound inequalities using the same techniques we used to solve linear inequalities. We solve each inequality separately and then consider the two solutions.

To solve a compound inequality with the word “and,” we look for all numbers that make both inequalities true. To solve a compound inequality with the word “or,” we look for all numbers that make either inequality true.

Let’s start with the compound inequalities with “and.” Our solution will be the numbers that are solutions to both inequalities known as the intersection of the two inequalities. Consider the intersection of two streets—the part where the streets overlap—belongs to both streets.

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To find the solution of an “and” compound inequality, we look at the graphs of each inequality and then find the numbers that belong to both graphs—where the graphs overlap.

For the compound inequality (x>−3) and (xleq 2), we graph each inequality. We then look for where the graphs “overlap”. The numbers that are shaded on both graphs, will be shaded on the graph of the solution of the compound inequality. See Figure (PageIndex{1}).

We can see that the numbers between (−3) and (2) are shaded on both of the first two graphs. They will then be shaded on the solution graph.

The number (−3) is not shaded on the first graph and so since it is not shaded on both graphs, it is not included on the solution graph.

The number two is shaded on both the first and second graphs. Therefore, it is be shaded on the solution graph.

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This is how we will show our solution in the next examples.

Sometimes we have a compound inequality that can be written more concisely. For example, (a<x) and (x<b) can be written simply as (a<x<b) and then we call it a double inequality. The two forms are equivalent.

To solve a double inequality we perform the same operation on all three “parts” of the double inequality with the goal of isolating the variable in the center.

When written as a double inequality, (1leq x<5), it is easy to see that the solutions are the numbers caught between one and five, including one, but not five. We can then graph the solution immediately as we did above.

Another way to graph the solution of (1leq x<5) is to graph both the solution of (xgeq 1) and the solution of (x<5). We would then find the numbers that make both inequalities true as we did in previous examples.

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