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

In this article, you will learn how to solve and write two-step inequalities, which are inequalities that involve two operations, such as addition and multiplication. You will also find some examples and practice problems from lesson 7 homework.

What are Two-Step Inequalities?

An inequality is a mathematical statement that compares two expressions using symbols like , ≤, or ≥. For example, 3x + 5 < 17 is an inequality that says that 3 times a number plus 5 is less than 17.

A two-step inequality is an inequality that requires two steps to solve. For example, to solve 3x + 5 < 17, you need to subtract 5 from both sides and then divide by 3. The solution is x < 4, which means that any number less than 4 makes the inequality true.

How to Solve Two-Step Inequalities?

To solve a two-step inequality, you need to follow these steps:

1. Use inverse operations to isolate the variable on one side of the inequality. Inverse operations are operations that undo each other, such as addition and subtraction, or multiplication and division.
2. When you perform an operation on one side of the inequality, you must do the same operation on the other side to keep the inequality balanced.
3. If you multiply or divide by a negative number, you must reverse the direction of the inequality symbol. This is because multiplying or dividing by a negative number changes the order of the numbers on a number line.
4. Check your solution by plugging it into the original inequality and making sure it makes the statement true.

Examples of Solving Two-Step Inequalities

Here are some examples of solving two-step inequalities from lesson 7 homework practice:

1. Solve and write the solution in interval notation: 2x – 7 > 9
2. Add 7 to both sides: 2x > 16
3. Divide by 2: x > 8
4. The solution in interval notation is (8, ∞), which means all numbers greater than 8.
5. Check: Plug in a number greater than 8, such as 10, into the original inequality: 2(10) – 7 > 9. Simplify: 20 – 7 > 9. Simplify: 13 > 9. This is true, so the solution is correct.

2. Solve and write the solution in interval notation: -4y + 12 ≤ -8
3. Subtract 12 from both sides: -4y ≤ -20
4. Divide by -4 and reverse the inequality symbol: y ≥ 5
5. The solution in interval notation is [5, ∞), which means all numbers greater than or equal to 5.
6. Check: Plug in a number greater than or equal to 5, such as 6, into the original inequality: -4(6) + 12 ≤ -8. Simplify: -24 + 12 ≤ -8. Simplify: -12 ≤ -8. This is true, so the solution is correct.

How to Graph Two-Step Inequalities?

To graph a two-step inequality, you need to follow these steps:

1. Write the inequality in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. To do this, you may need to use the same steps as solving a two-step inequality.
2. Graph the line y = mx + b using the slope and the y-intercept. If the inequality symbol is , use a dashed line. If the inequality symbol is ≤ or ≥, use a solid line.
3. Shade the region above or below the line depending on the inequality symbol. If the inequality symbol is or ≥, shade above the line.
4. Check your graph by picking a point in the shaded region and plugging it into the original inequality. If it makes the statement true, your graph is correct.

Examples of Graphing Two-Step Inequalities

Here are some examples of graphing two-step inequalities from lesson 7 homework practice:

1. Graph and write the solution in interval notation: y > 2x – 3
2. The inequality is already in slope-intercept form, so we can use m = 2 and b = -3 to graph the line.
3. Since the inequality symbol is >, we use a dashed line.
4. We shade above the line because y is greater than something.
5. The solution in interval notation is {y | y > 2x – 3}, which means all values of y that are greater than 2x – 3.
6. Check: Pick a point in the shaded region, such as (0, 0), and plug it into the original inequality: 0 > 2(0) – 3. Simplify: 0 > -3. This is true, so the graph is correct.

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2. Graph and write the solution in interval notation: y ≤ -4x + 12
3. The inequality is already in slope-intercept form, so we can use m = -4 and b = 12 to graph the line.
4. Since the inequality symbol is ≤, we use a solid line.
5. We shade below the line because y is less than or equal to something.
6. The solution in interval notation is {y | y ≤ -4x + 12}, which means all values of y that are less than or equal to -4x + 12.
7. Check: Pick a point in the shaded region, such as (1, 5), and plug it into the original inequality: 5 ≤ -4(1) + 12. Simplify: 5 ≤ 8. This is true, so the graph is correct.

Conclusion

In this article, you have learned how to solve and write two-step inequalities, which are inequalities that involve two operations. You have also learned how to graph two-step inequalities on a coordinate plane and write the solution in interval notation. You have practiced these skills with some examples from lesson 7 homework. Solving and writing two-step inequalities is an important skill that can help you model and analyze real-world situations involving quantities that vary or have constraints.

Quiz Question

Here is a quiz question based on the article to test your understanding of two-step inequalities:

Solve and graph the following two-step inequality and write the solution in interval notation: 3x – 5 < 7

A) x < 4; shade below the dashed line y = 3x – 5; {x | x < 4}

B) x < 4; shade above the dashed line y = 3x – 5; {x | x < 4}

C) x > 4; shade below the solid line y = 3x – 5; {x | x > 4}

D) x > 4; shade above the solid line y = 3x – 5; {x | x > 4}

The correct answer is A) x < 4; shade below the dashed line y = 3x – 5; {x | x < 4}. To solve the inequality, you need to add 5 to both sides and then divide by 3. To graph the inequality, you need to use a dashed line because the inequality symbol is < and shade below the line because y is less than something.

How to Write Two-Step Inequalities?

Sometimes, you may need to write a two-step inequality from a word problem or a real-world situation. To do this, you need to follow these steps:

1. Identify the variable and assign it a letter, such as x.
2. Translate the words into mathematical expressions using the variable and the inequality symbols.
3. Simplify the expressions if possible.
4. Check your inequality by plugging in some values and making sure it matches the situation.

Examples of Writing Two-Step Inequalities

Here are some examples of writing two-step inequalities from lesson 7 homework practice:

1. A plumber charges $50 for a service call plus $75 per hour of work. Write an inequality for the total cost, C, of a service call that lasts at most 3 hours.
2. Let x be the number of hours of work. The total cost is 50 + 75x.
3. The service call lasts at most 3 hours, which means x is less than or equal to 3.
4. The inequality is C ≤ 50 + 75x.
5. Check: Plug in some values for x and C and make sure they make sense. For example, if x = 2, then C ≤ 50 + 75(2) = 200. This means that the total cost is at most $200 for a service call that lasts 2 hours. This matches the situation.

2. A car rental company charges $25 per day plus $0.15 per mile for renting a car. Write an inequality for the number of miles, m, that a person can drive in one day if they have at most $40 to spend.
3. The total cost is 25 + 0.15m.
4. The total cost is at most $40, which means 25 + 0.15m is less than or equal to 40.
5. The inequality is 25 + 0.15m ≤ 40.
6. Check: Plug in some values for m and make sure they make sense. For example, if m = 100, then 25 + 0.15(100) = 40. This means that the person can drive exactly 100 miles in one day with $40. This matches the situation.

How to Solve and Write Compound Inequalities?

A compound inequality is an inequality that combines two simple inequalities using the words “and” or “or”. For example, 2x – 3 > 5 and x + 4 < 8 is a compound inequality. To solve and write a compound inequality, you need to follow these steps:

1. Solve each simple inequality separately using the same steps as solving a two-step inequality.
2. If the compound inequality uses the word “and”, then find the intersection of the two solutions. This means finding the values that make both inequalities true.
3. If the compound inequality uses the word “or”, then find the union of the two solutions. This means finding the values that make either inequality true.
4. Write the solution in interval notation or set-builder notation.
5. Check your solution by plugging in some values and making sure they make the compound inequality true.

Examples of Solving and Writing Compound Inequalities

Here are some examples of solving and writing compound inequalities from lesson 7 homework practice:

1. Solve and write the solution in interval notation: 2x – 3 > 5 and x + 4 < 8
2. Solve each simple inequality separately: 2x – 3 > 5 –> x > 4; x + 4 x < 4
3. The compound inequality uses the word “and”, so we find the intersection of the two solutions. There are no values that make both inequalities true, so the solution is the empty set.
4. The solution in interval notation is ∅.
5. Check: There are no values to plug in, so we can’t check the solution.

2. Solve and write the solution in interval notation: -3x + 5 ≤ -7 or x – 2 ≥ 6
3. Solve each simple inequality separately: -3x + 5 ≤ -7 –> x ≥ 4; x – 2 ≥ 6 –> x ≥ 8
4. The compound inequality uses the word “or”, so we find the union of the two solutions. The values that make either inequality true are x ≥ 4.
5. The solution in interval notation is [4, ∞).
6. Check: Plug in some values for x that are greater than or equal to 4 and make sure they make either inequality true. For example, if x = 5, then -3(5) + 5 ≤ -7 and 5 – 2 ≥ 6. Simplify: -10 ≤ -7 and 3 ≥ 6. The first inequality is true, so the solution is correct.

Conclusion

In this article, you have learned how to solve and write two-step inequalities and compound inequalities. You have also learned how to graph two-step inequalities and write the solution in interval notation. You have practiced these skills with some examples from lesson 7 homework. Solving and writing two-step inequalities and compound inequalities is an important skill that can help you model and analyze real-world situations involving quantities that vary or have constraints.

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