Solve the Inequality for x x/-1> 2/5
The given question is asking for the solution to a mathematical inequality. The inequality in question is "x/-1 > 2/5", and you are supposed to find the set of values that the variable x can take in order to make the inequality true. Essentially, you need to manipulate the inequality to isolate x and determine its range of possible values based on the given condition. Remember that when solving inequalities, certain operations, like multiplying or dividing by a negative number, require you to flip the direction of the inequality sign.
$\frac{x}{- 1} > \frac{2}{5}$
Answer
Solution:
Step 1:
Eliminate the negative denominator by multiplying both sides of $\frac{x}{-1} > \frac{2}{5}$ by $-1$ to get $-x > \frac{2}{5}$.
Step 2:
Invert the inequality sign and divide both sides by $-1$ to isolate $x$.
Step 2.1:
Apply the rule that the inequality sign reverses when both sides are multiplied or divided by a negative number: $-x > \frac{2}{5}$ becomes $x < -\frac{2}{5}$ after division by $-1$.
Step 2.2:
Simplify the expression on the left.
Step 2.2.1:
Recognize that dividing a negative by a negative gives a positive: $\frac{-x}{-1} = x$.
Step 2.2.2:
Simplify the division of $x$ by $1$: $x = x$.
Step 2.3:
Simplify the expression on the right.
Step 2.3.1:
Multiply the negative outside the fraction by the numerator: $x < -1 \cdot \frac{2}{5}$.
Step 2.3.2:
Express the multiplication of $-1$ and $\frac{2}{5}$ as a single fraction: $x < -\frac{2}{5}$.
Step 3:
Express the solution in different forms.
Inequality Form: $x < -\frac{2}{5}$ Interval Notation: $(-\infty, -\frac{2}{5})$
Knowledge Notes:
To solve the inequality $\frac{x}{-1} > \frac{2}{5}$, we must manipulate the inequality to isolate $x$ while adhering to the rules of inequalities. Here are the relevant knowledge points:
Multiplying or Dividing by a Negative Number: When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. This is because the order of the numbers on the number line changes when multiplied by a negative number.
Simplification: Dividing a number by 1 does not change its value, so $\frac{x}{1} = x$. Similarly, multiplying a number by $-1$ changes its sign, so $-1 \cdot \frac{2}{5} = -\frac{2}{5}$.
Interval Notation: This is a way of writing the set of all numbers between two endpoints. The notation $(-\infty, -\frac{2}{5})$ represents all numbers less than $-\frac{2}{5}$.
Inequalities: These are mathematical expressions that compare two values, showing that one value is less than, greater than, less than or equal to, or greater than or equal to another value.
Latex Formatting: In the solution, Latex is used to format mathematical expressions, ensuring that they are clearly presented and easy to read. For example, $\frac{x}{-1}$ is rendered in Latex to display the fraction properly.
