Inequalities and their solution sets

Inequalities show how two amounts compare when they are not equal. Here are the four symbols you will use. The greater than symbol > means the left side is bigger. The less than symbol < means the left side is smaller. When a small line sits under the symbol, it adds the words or equal to. So greater than or equal to means the number itself is allowed, and less than or equal to works the same way.

You can draw the solutions of an inequality on a number line. Think about x > 4. This means x can be 5, 6, 7, or any bigger number, but 4 itself is not allowed. Draw an open circle on 4 to show that 4 is left out, then shade the line to the right, where the bigger numbers live.

Now try x is less than or equal to 4. This time 4 itself is allowed, so you draw a closed, filled-in circle on 4. Since the solutions are 4 and the smaller numbers, you shade to the left.

Strict symbols like > and < leave the number out, so they get an open circle. Symbols with or equal to include the number, so they get a closed circle. Then the small side of the symbol points toward the smaller numbers on your shading.

Linear inequalities pop up all the time in real life. A linear equation gives you exactly one answer when you solve it. A linear inequality gives you a whole crowd of answers.

Look at these everyday statements and how they turn into inequalities. If a trip will take at least 30 minutes, the waiting time x is 30 or more, so you write x is greater than or equal to 30. Thirty minutes works, and so do 45 or 60. If you would not pay more than 6 dollars for a sandwich, then the price x is 6 or less, so you write x is less than or equal to 6. Six dollars works, and so does anything cheaper.

The tricky part is translating the words. At least means the number itself is allowed, so use or equal to. No more than and at most mean the same thing, the number or anything below it. More than and less than leave the number out.

Once you have the inequality, you solve it just like an equation. Use inverse operations, the opposite moves, to peel away the numbers around x. The only surprise is that an inequality has many answers, not just one. That is why we draw the answers on a number line instead of circling a single number.

This standard is about solving linear equations and inequalities in one variable, which means finding every value that makes the statement true.

An equation like 2x + 3 = 11 has exactly one answer. An inequality like 2x + 3 < 11 has many answers, because lots of numbers make it true. The good news is that you solve both the same way, using inverse operations. That means doing the opposite move to undo each number wrapped around x.

First undo the +3 by subtracting 3 from both sides, which gives 2x < 8. Then undo the times 2 by dividing both sides by 2, which gives x < 4. Every number smaller than 4 is a solution.

You can check your work the same way too. Pick a number from your answer and plug it back in. Try x = 0 in 2x + 3 < 11. That gives 3 < 11, which is true. Now try a number that should not work, like x = 5. That gives 13 < 11, which is false, so 5 really is not a solution. Checking both ways like this shows your solution set is right.