Lesson 7 Homework Practice Solve And Write Two Step Inequalities Answers

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Lesson 7 Homework Practice: Solve and Write Two-Step Inequalities

In this lesson, you will learn how to solve and write two-step inequalities. A two-step inequality is an inequality that requires two steps to solve. For example, consider the inequality -1/4 (8n-12) â¥ â2nâ1. To solve this inequality, you need to do the following steps:

Distribute the -1/4 on the left side. -1/4 (8n-12) â¥ â2nâ1 becomes -2n+3 â¥ â2nâ1.

Add 2n to both sides. -2n+3 â¥ â2nâ1 becomes 3 â¥ â1.

The solution is 3 â¥ â1, which means that any value of n will make the inequality true. This is called an identity, because it is always true.

To write a two-step inequality, you need to reverse the process. For example, suppose you want to write a two-step inequality that has the solution n > 5. You can do the following steps:

Subtract a number from both sides. For example, subtract 3 from both sides. n > 5 becomes n - 3 > 2.

Multiply or divide both sides by a number. For example, multiply both sides by -2. n - 3 > 2 becomes -2(n - 3) < -4. Note that when you multiply or divide by a negative number, you need to flip the inequality sign.

The two-step inequality is -2(n - 3) < -4, which has the same solution as n > 5.

Here are some practice problems for you to try:

Solve and write the solution in interval notation: 3x + 5 â¤ 11

Solve and write the solution in interval notation: -4y + 8 > 0

Write a two-step inequality that has the solution x â¤ -2

Write a two-step inequality that has the solution y > 3/4

You can check your answers using this website[^2^]. Good luck!Here are some more details about two-step inequalities:

A two-step inequality can have one of four types of solutions:

An identity: This means that the inequality is always true, no matter what value you choose for the variable. For example, 2x + 4 â¥ 2x + 3 is an identity, because any value of x will make the left side greater than or equal to the right side.

A contradiction: This means that the inequality is never true, no matter what value you choose for the variable. For example, 3x - 5 < 3x - 6 is a contradiction, because any value of x will make the left side greater than or equal to the right side.

A single value: This means that the inequality is only true for one specific value of the variable. For example, 4x - 8 = 0 is only true when x = 2.

An interval: This means that the inequality is true for a range of values of the variable. For example, x + 2 > -1 is true when x > -3.

To write the solution of an inequality in interval notation, you need to use brackets and parentheses to indicate whether the endpoints are included or excluded. For example, x > -3 can be written as (-3, â), which means that x can be any number greater than -3 but not equal to -3. The symbol â means infinity, which means that there is no upper limit for x. On the other hand, x â¥ -3 can be written as [-3, â), which means that x can be any number greater than or equal to -3.

Here are some examples of how to write solutions in interval notation:

InequalitySolutionInterval Notation

-2x + 6 â¤ 10x â¥ -2[-2, â)

5y - 15 > 10y > 5(5, â)

-3z + 9 = 0z = 3{3}

x + 4 < x + 5identity(-â, â)

x - 7 > x - 6contradictionâ

The symbol â means the empty set, which means that there is no solution. 061ffe29dd