Problem 7 For Exercises 5 to $13,$ solve... [FREE SOLUTION]

Chapter 14: Problem 7

For Exercises 5 to $13,$ solve and write the answer in interval notation. $$2 x-8>4 x$$

Short Answer

Expert verified

The solution is $-\infty, -4)$.

Step by step solution

Understand the Task at Hand

Given the inequality $2x - 8 > 4x$, the task is to solve for $x$, which means rearranging the inequality to make $x$ the subject.

Rearrange the inequality

Firstly, subtract $2x$ from both sides to keep the inequality balanced. That gives: $-8 > 2x$. Then further divide by 2 on both sides to look for the value of $x$. You will get: $-4 > x$. Rewriting this in a more standard form gives: $x< -4$.

Convert to Interval Notation

Now, write the solution to the inequality in interval notation. This notation describes an interval of values that would satisfy the inequality. In the case of $x< -4$, the solution in interval notation is: $-\infty, -4)$.

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Key Concepts

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

Interval Notation

Interval notation is a way of representing a range of values along a number line. It is particularly useful for expressing solutions to inequalities. For the inequality solution where we have variables less than a certain number, such as $x < -4$, we use open intervals. Here's a quick guide on how this works:

Use parentheses $()$ to signify that an endpoint is not included in the interval (an open interval).
Use brackets $[]$ if the endpoint is included (a closed interval).

For our inequality $x < -4$, the interval notation is $(-\infty, -4)$. This indicates all numbers from negative infinity up to, but not including, -4. Remember, infinity and negative infinity always use parentheses because we can't "reach" infinity.

Algebraic Manipulation

Algebraic manipulation is a crucial skill when solving inequalities. It involves rearranging and simplifying expressions to isolate the variable of interest. In the original problem, $2x - 8 > 4x$, let's break down the process:

Firstly, ensure the inequality remains balanced by performing the same operation on both sides. We start by eliminating $2x$ from the left side:\[2x - 8 > 4x \Rightarrow 2x - 8 - 2x > 4x - 2x\]This simplifies to:\[-8 > 2x\]Next, divide each side by 2 to solve for $x$:\[-4 > x\]
By rearranging, we find $x < -4$. Each step in algebraic manipulation helps in isolating $x$ so we can describe its possible values.

Linear Inequalities

Linear inequalities, like $2x - 8 > 4x$, are similar to linear equations, but instead of an equal sign, they use inequality symbols. These symbols include $<$, $>$, $\leq$, and $\geq$.

When solving these inequalities, the goal is to find all values of $x$ that make the inequality true. Here are key steps:

Isolate the variable on one side using basic algebraic operations.
When multiplying or dividing by a negative number, flip the inequality sign.
Finally, express the solution in interval notation.

In the example, through proper manipulation, we reached $x < -4$. This tells us that any number less than -4 will satisfy the inequality. Linear inequalities help us understand and express ranges of solutions that aren't tied to one specific value.

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91影视

For Exercises 5 to \(13,\) solve and write the answer in interval notation. $$2 x-8>4 x$$

Short Answer

Step by step solution

Understand the Task at Hand

Rearrange the inequality

Convert to Interval Notation

Key Concepts

Interval Notation

Algebraic Manipulation

Linear Inequalities

One App. One Place for Learning.

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