Solve The Following Two-step Inequalities:1) $\frac{x+9}{14} \ \textgreater \ 1$ □ 2) $40 \geq 8(12+x$\] □ 3) $6 \ \textless \ 6(-2+x$\] □ 4) $-2x - 7 \ \textgreater \ -11$ □ 5) $9 - \frac{x}{4} \

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In this article, we will focus on solving two-step inequalities. Two-step inequalities are inequalities that require two steps to solve. These inequalities involve variables and constants, and the goal is to isolate the variable on one side of the inequality sign. We will use algebraic methods to solve these inequalities.

Step 1: Solve the Inequality $\frac{x+9}{14} \ \textgreater \ 1$

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To solve this inequality, we need to isolate the variable x. We can start by multiplying both sides of the inequality by 14, which is the denominator of the fraction.

\frac{x+9}{14} \  \textgreater \  1
\Rightarrow (x+9) \  \textgreater \  14

Next, we can subtract 9 from both sides of the inequality to isolate the variable x.

(x+9) \  \textgreater \  14
\Rightarrow x \  \textgreater \  5

Therefore, the solution to the inequality $\frac{x+9}{14} \ \textgreater \ 1$ is $x \ \textgreater \ 5$.

Step 2: Solve the Inequality $40 \geq 8(12+x)$

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To solve this inequality, we need to isolate the variable x. We can start by distributing the 8 to the terms inside the parentheses.

40 \geq 8(12+x)
\Rightarrow 40 \geq 96 + 8x

Next, we can subtract 96 from both sides of the inequality to isolate the term with the variable x.

40 \geq 96 + 8x
\Rightarrow -56 \geq 8x

Finally, we can divide both sides of the inequality by 8 to isolate the variable x.

-56 \geq 8x
\Rightarrow -7 \geq x

Therefore, the solution to the inequality $40 \geq 8(12+x)$ is $x \leq -7$.

Step 3: Solve the Inequality $6 \ \textless \ 6(-2+x)$

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To solve this inequality, we need to isolate the variable x. We can start by distributing the 6 to the terms inside the parentheses.

6 \  \textless \  6(-2+x)
\Rightarrow 6 \  \textless \  -12 + 6x

Next, we can add 12 to both sides of the inequality to isolate the term with the variable x.

6 \  \textless \  -12 + 6x
\Rightarrow 18 \  \textless \  6x

Finally, we can divide both sides of the inequality by 6 to isolate the variable x.

18 \  \textless \  6x
\Rightarrow 3 \  \textless \  x

Therefore, the solution to the inequality $6 \ \textless \ 6(-2+x)$ is $x \ \textgreater \ 3$.

Step 4: Solve the Inequality $-2x - 7 \ \textgreater \ -11$

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To solve this inequality, we need to isolate the variable x. We can start by adding 7 to both sides of the inequality to isolate the term with the variable x.

-2x - 7 \  \textgreater \  -11
\Rightarrow -2x \  \textgreater \  -4

Next, we can divide both sides of the inequality by -2 to isolate the variable x. Note that when we divide by a negative number, the inequality sign is reversed.

-2x \  \textgreater \  -4
\Rightarrow x \  \textless \  2

Therefore, the solution to the inequality $-2x - 7 \ \textgreater \ -11$ is $x \ \textless \ 2$.

Step 5: Solve the Inequality $9 - \frac{x}{4} \ \textless \ 3$

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To solve this inequality, we need to isolate the variable x. We can start by subtracting 9 from both sides of the inequality to isolate the term with the variable x.

9 - \frac{x}{4} \  \textless \  3
\Rightarrow -\frac{x}{4} \  \textless \  -6

Next, we can multiply both sides of the inequality by -4 to isolate the variable x. Note that when we multiply by a negative number, the inequality sign is reversed.

-\frac{x}{4} \  \textless \  -6
\Rightarrow x \  \textgreater \  24

Therefore, the solution to the inequality $9 - \frac{x}{4} \ \textless \ 3$ is $x \ \textgreater \ 24$.

Conclusion

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In this article, we have solved five two-step inequalities. We have used algebraic methods to isolate the variable x in each inequality. The solutions to the inequalities are as follows:

• $\frac{x+9}{14} \ \textgreater \ 1$: $x \ \textgreater \ 5$
• $40 \geq 8(12+x)$: $x \leq -7$
• $6 \ \textless \ 6(-2+x)$: $x \ \textgreater \ 3$
• $-2x - 7 \ \textgreater \ -11$: $x \ \textless \ 2$
• $9 - \frac{x}{4} \ \textless \ 3$: $x \ \textgreater \ 24$

We hope that this article has been helpful in understanding how to solve two-step inequalities.

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In this article, we will answer some frequently asked questions about two-step inequalities. Two-step inequalities are inequalities that require two steps to solve. These inequalities involve variables and constants, and the goal is to isolate the variable on one side of the inequality sign.

Q: What is a two-step inequality?

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A two-step inequality is an inequality that requires two steps to solve. These inequalities involve variables and constants, and the goal is to isolate the variable on one side of the inequality sign.

Q: How do I solve a two-step inequality?

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

1. Distribute any numbers that are being multiplied to the terms inside the parentheses.
2. Combine like terms on each side of the inequality.
3. Isolate the variable on one side of the inequality sign.

Q: What is the difference between a two-step inequality and a one-step inequality?

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A one-step inequality is an inequality that can be solved in one step. For example, the inequality $x + 3 > 5$ can be solved in one step by subtracting 3 from both sides of the inequality.

A two-step inequality, on the other hand, requires two steps to solve. For example, the inequality $\frac{x+9}{14} \ \textgreater \ 1$ requires two steps to solve: first, we need to multiply both sides of the inequality by 14, and then we need to subtract 9 from both sides of the inequality.

Q: How do I know which operation to perform first when solving a two-step inequality?

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When solving a two-step inequality, you need to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate any expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: What if I have a fraction in a two-step inequality?

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If you have a fraction in a two-step inequality, you can multiply both sides of the inequality by the denominator of the fraction to eliminate the fraction. For example, if you have the inequality $\frac{x+9}{14} \ \textgreater \ 1$, you can multiply both sides of the inequality by 14 to eliminate the fraction.

Q: Can I use a calculator to solve a two-step inequality?

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Yes, you can use a calculator to solve a two-step inequality. However, you need to make sure that you are using the correct operation and that you are following the order of operations (PEMDAS).

Q: How do I check my solution to a two-step inequality?

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To check your solution to a two-step inequality, you need to plug your solution back into the original inequality and make sure that it is true. For example, if you have the inequality $x \ \textgreater \ 5$ and you plug in $x = 6$, you should get a true statement.

Conclusion

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In this article, we have answered some frequently asked questions about two-step inequalities. We hope that this article has been helpful in understanding how to solve two-step inequalities and how to use a calculator to solve them.

Additional Resources

• Two-Step Inequality Examples
• Two-Step Inequality Practice Problems

Two-Step Inequality Examples

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Here are some examples of two-step inequalities:

• $\frac{x+9}{14} \ \textgreater \ 1$
• $40 \geq 8(12+x)$
• $6 \ \textless \ 6(-2+x)$
• $-2x - 7 \ \textgreater \ -11$
• $9 - \frac{x}{4} \ \textless \ 3$

Two-Step Inequality Practice Problems

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Here are some practice problems for two-step inequalities:

• Solve the inequality $\frac{x-3}{5} \ \textgreater \ 2$.
• Solve the inequality $24 \geq 4(6+x)$.
• Solve the inequality $3 \ \textless \ 3(-1+x)$.
• Solve the inequality $-3x - 2 \ \textgreater \ -8$.
• Solve the inequality $7 - \frac{x}{3} \ \textless \ 2$.

We hope that these examples and practice problems have been helpful in understanding how to solve two-step inequalities.