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Solving Linear Inequalities

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Let's look at our objectives for today. We'll start by defining what is an inequality. We'll then explore different operations used in inequality statements. We'll compare solving an inequality with solving an equation. And finally, we'll do some examples of solving an inequality.

Let's start by looking at what an inequality is. An inequality is a mathematical statement that two quantities are not equal in value. We use inequality symbols to show that one quantity is greater than or less than another quantity.

This is the inequality symbol which means less than. For example, we can write 3 is less than 5. This is the inequality symbol which means greater than. Example, we can write 7 is greater than 5. This inequality symbol means less than or equal to. For example, we can write 2 is less than or equal to 4.

And finally, this inequality symbol means greater than or equal to. For example, we can write 6 is greater than or equal to 4. Inequalities are used in statistics, business, economics, and optimization when comparing values to each other.

Now let's explore how we use operations in an inequality. Just like with equations, we can add or subtract a value on both sides of an inequality and keep the inequality statement true. For example, if we have the inequality 3 is less than 5, and we add 2 to both sides, we have 3 plus 2 is less than 5 plus 2, which simplifies to 5 is less than 7, which is still a true statement.

We can also multiply or divide by a positive number on both sides and keep the inequality statement true. For example, if we have the inequality 6 is greater than or equal to 4, and we multiply by 3 on both sides, we have 3 times 6 is greater than or equal to 3 times 4, which simplifies to 18 is greater than or equal to 12, which is still true.

However, if we multiply or divide by a negative number on both sides of the inequality, the statement becomes untrue until we flip or reverse the inequality sign. For example, if we have the inequality 6 is greater than or equal to 4, and we now multiply by a negative 3 on both sides, we have negative 3 times 6 is greater than or equal to negative 3 times 4, which simplifies to negative 18 is greater than or equal to negative 12, which is untrue, until we flip the sign. When we flip the sign, we have negative 18 is less than or equal to negative 12, which is a true statement.

Now let's compare solving an equation to solving an inequality. Recall that in solving an equation, we isolate the variable using inverse operations. In order to isolate the variable, inverse operations are used to get all terms involving the variable on one side of the equation and all other terms to the other side of the equation.

The process for solving inequality follows the same rules. We use inverse operations to isolate the variable, and what is done on one side of the inequality must be done to the other. The important difference is that when we multiply or divide by a negative number, we have to flip the inequality sign.

Now let's do some examples solving inequalities. Suppose Marcus has accepted a job selling cellphones. He will be paid $1,500 plus 15% of his sales each month. He's needs to earn at least $2,430 per month to pay his bills. For what amount of sales will Marcus be able to pay his bills?

We start with $1,500. And we know that Marcus earns an additional 15%, or 0.15, of the total sales, which is x. This has to be at least or greater than or equal to 2,430. So our inequality becomes 1,500 plus 0.15x is greater than or equal to 2,430.

To solve this inequality, we start by subtracting 1,500 on both sides to isolate our x variable. This gives us 0.15x is greater than or equal to 930. We then divide both sides by 0.15, which gives us x is greater than or equal to 6,200. So Marcus' total sales must be at least $6,200 in order for him to pay his bills.

We can show this solution on a number line and use it to check our answer. If we pick a value in the highlighted range below it should satisfy our original inequality. Let's pick the value 6,500. Substituting it back into our inequality, we have 1,500 plus 0.15 times 6,500 is greater than or equal to 2,430. Simplifying starting with multiplication on the left side gives us 1,500 plus 975 is greater than or equal to 2,430. We then add 1,500 plus 975, which gives us 2,475 greater than or equal to 2,430, which is a true statement.

Here's a second example. Wei has a job paying $25,000 a year and expects a raise of $1,000 each year. Jaime has a job paying $19,000 and expects to receive a raise of $1,500 each year. When will Jaime be making more than Wei?

We know that Wei starts at $25,000 and gets a raise of $1,000 for each year t. Jaime starts at 19,000 and gets a raise of 1,500 for each year t. We want to know when Jaime will make more than Wei, so our inequality becomes 25,000 plus 1,000t is less than 19,000 plus 1,500t.

The left side of the inequality represents the amount that Wei makes. And the right side of the inequality represents how much Jaime makes. To solve, we need to isolate our variable on one side, so we start by subtracting 1,500t from both sides of the inequality. This gives us 25,000 minus 500t is less than 19,000.

We then subtract 25,000 from both sides of the inequality, which gives us negative 500t is less than negative 6,000. Finally, we divide both sides of the inequality by negative 500. Since we're dividing by a negative number, we need to flip the inequality sign. So we now have t is greater than 12. So Jaime will be making more than Wei after 12 years.

We can show the solution on a number line shown below. If we pick a value in the highlighted range it should satisfy our original inequality. Let's pick the value 15. Substituting 15 back into our original inequality for t gives us 25,000 plus 1,000 times 15 is less than 19,000 plus 1,500 times 15.

Simplifying with multiplication on both sides of the inequality gives us 25,000 plus 15,000 is less than 19,000 plus 22,500. This gives us 40,000 is less than 41,500, which is a true statement.

Let's review our important points from today. Make sure you get them in your notes so you can refer to them later. The inequality symbols include less than, less than or equal to, greater than, and greater than or equal to. When solving an inequality, we follow the same steps as in solving an equation to isolate the variable.

However, it's important to remember that when we multiply or divide by a negative number, the inequality symbol must be flipped in order to maintain a true statement. And finally, a number line can be used to visually show the range of solutions to an inequality problem.

So I hope that these key points and examples helped you understand a little bit more about solving linear inequalities. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks for watching.