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Chapter 3: Problem 24

$$ \text { For Problems 23-60, solve each inequality. (Objective 1) } $$ $$ x+7>-15 $$

Short Answer

Expert verified
The solution is \( x > -22 \).

Step by step solution

01

Isolate the Variable

The goal is to solve for \( x \) in the inequality \( x + 7 > -15 \). Begin by isolating \( x \) on one side of the inequality. To do this, subtract 7 from both sides of the inequality: \[ x + 7 - 7 > -15 - 7 \] This simplifies to: \[ x > -22 \]
02

Interpret the Solution

The inequality \( x > -22 \) means that \( x \) can be any number greater than \(-22\). This represents all the real numbers on the number line that are to the right of \(-22\).
03

Verify the Solution

To check the solution, pick a value for \( x \) greater than \(-22\), and substitute back into the original inequality to verify. For example, if \( x = 0 \): \[ 0 + 7 > -15 \] \[ 7 > -15 \] Since the statement is true, the solution is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating the Variable
When solving inequalities like \( x + 7 > -15 \), the first step is to isolate the variable. This means you want \( x \) on one side of the inequality and all the constants on the other. To achieve this, perform the same operation to both sides of the inequality. By subtracting 7 from both sides, we ensure balance is maintained. The process looks like this: \( x + 7 - 7 > -15 - 7 \).

Once you simplify, you end up with \( x > -22 \). Now, \( x \) is isolated, and you can clearly see the range of possible values for \( x \). Remember, isolating the variable helps make the inequality easier to read and understand.
Interpreting the Solution
After isolating the variable, the next goal is to interpret what the solution means. In this case, \( x > -22 \) indicates that \( x \) can take any value greater than \(-22\). Visualizing this on a number line can help tremendously.

Imagine all the points to the right of \(-22\) as possibilities for \( x \). In other words, any number larger than \(-22\) satisfies this inequality. This includes integers, fractions, and decimals as long as they are greater than \(-22\).
  • For example, values such as -21, 0, and even 1000 are valid options.
By understanding the result, you're preparing yourself to apply this solution in different contexts.
Verifying the Solution
To be confident in your solution, it鈥檚 crucial to verify it. Verification involves checking whether your solution holds true in the original inequality. Start by selecting a specific value for \( x \) greater than \(-22\). Let's take \( x = 0 \) as an example.

When substituting \( x = 0 \) back into the original inequality, you compute \( 0 + 7 > -15 \). This simplifies to \( 7 > -15 \), which is true. Hence, selecting \( x = 0 \) confirms our solution is correct.
  • Try picking different numbers, such as \( x = -21 \) or \( x = 1 \), to further support your solution.
This step is critical as it ensures the accuracy of your solution and builds confidence in your mathematical reasoning skills.

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