A description of different types of inequalities follows. A strict inequality is a relation that holds between two values when they are different. To compare the size of the values, there are two types of relations:. The above relations can be demonstrated on a number line. Recall that the values on a number line increase as you move to the right. In contrast to strict inequalities, there are two types of inequality relations that are not strict:. In addition to showing relationships between integers, inequalities can be used to show relationships between variables and integers.
For a visualization of this, see the number line below:. Note that an open circle is used if the inequality is strict i. Likewise, inequalities can be used to demonstrate relationships between different expressions. One useful application of inequalities such as these is in problems that involve maximum or minimum values. Jared has a boat with a maximum weight limit of 2, pounds. He wants to take as many of his friends as possible onto the boat, and he guesses that he and his friends weigh an average of pounds.
How many people can ride his boat at once? To see why this is so, consider the left side of the inequality. There are steps that can be followed to solve an inequality such as this one. For now, it is important simply to understand the meaning of such statements and cases in which they might be applicable.
As long as the same value is added or subtracted from both sides, the resulting inequality remains true. Take note that multiplying or dividing an inequality by a negative number changes the direction of the inequality. In other words, a greater-than symbol becomes a less-than symbol, and vice versa. This statement also holds true. This demonstrates how crucial it is to change the direction of the greater-than or less-than symbol when multiplying or dividing by a negative number.
Solving an inequality that includes a variable gives all of the possible values that the variable can take that make the inequality true. To solve an inequality means to transform it such that a variable is on one side of the symbol and a number or expression on the other side.
Often, multiple operations are often required to transform an inequality in this way. To see how the rules of addition and subtraction apply to solving inequalities, consider the following:. This means that we must also change the direction of the symbol:. It is not an actual number. Figure shows both the number line and the interval notation. The inequality means all numbers less than or equal to 1. There is no lower end to those numbers. We write in interval notation as.
Figure shows both the number line and interval notation. Inequalities, Number Lines, and Interval Notation Did you notice how the parenthesis or bracket in the interval notation matches the symbol at the endpoint of the arrow? These relationships are shown in Figure.
Graph on the number line and write in interval notation. The Subtraction and Addition Properties of Equality state that if two quantities are equal, when we add or subtract the same amount from both quantities, the results will be equal.
Similarly we could show that the inequality also stays the same for addition. We use these properties to solve inequalities, taking the same steps we used to solve equations. Solving the inequality , the steps would look like this:. Solve the inequality , graph the solution on the number line, and write the solution in interval notation. Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
Are there similar properties for inequalities? What happens to an inequality when we divide or multiply both sides by a constant?
Does the inequality stay the same when we divide or multiply by a negative number? When we divide or multiply an inequality by a positive number, the inequality sign stays the same.
When we divide or multiply an inequality by a negative number, the inequality sign reverses. Here are the Division and Multiplication Properties of Inequality for easy reference.
When we divide or multiply an inequality by a:. Solve each inequality, graph the solution on the number line, and write the solution in interval notation. Sometimes when solving an inequality, the variable ends up on the right. We can rewrite the inequality in reverse to get the variable to the left. Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but be sure to pay close attention during multiplication or division.
Just like some equations are identities and some are contradictions, inequalities may be identities or contradictions, too. We recognize these forms when we are left with only constants as we solve the inequality. If the result is a true statement, we have an identity. If the result is a false statement, we have a contradiction. To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality.
But others are not as obvious. It means 21 or more. Figure shows some common phrases that indicate inequalities. Translate and solve. Then write the solution in interval notation and graph on the number line. Graph Inequalities on the Number Line. In the following exercises, graph each inequality on the number line and write in interval notation. In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Solve Inequalities That Require Simplification. In the following exercises, translate and solve. Eight times z is smaller than. Ten times y is at most. Nineteen less than b is at most. Fifteen less than a is at least. Write this as an inequality. Fighter pilots The maximum height, h , of a fighter pilot is 77 inches.
Shopping The number of items, n , a shopper can have in the express check-out lane is at most 8. Explain why it is necessary to reverse the inequality when solving. What steps will you take to improve? In the following exercises, determine whether each number is a solution to the equation. In the following exercises, solve each equation using the Subtraction Property of Equality. In the following exercises, solve each equation using the Addition Property of Equality.
Solve Equations That Require Simplification. In the following exercises, translate each English sentence into an algebraic equation and then solve it. The sum of and is Four less than is In the following exercises, translate into an algebraic equation and solve. Her son is 3 years younger. How old is her son? Tan weighs pounds. Mark Mark 7, 6 6 gold badges 28 28 silver badges 63 63 bronze badges.
Then you want to prove that the real part of a complex number is less than the modulus. It holds since one side in a right triangle is always less than the hypotenuse. Add a comment. Active Oldest Votes. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown.
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