Solve the Inequality for x 5.3> x+2/5
The problem is asking to determine the range of values for the variable x that satisfy the given inequality. Specifically, you are to find all possible values of x such that when you add 2/5 to x, the result will be less than 5.3. In other words, you need to manipulate the inequality to isolate x on one side and compare it with a numerical value on the other, thus defining the solution set for the inequality.
$5.3 > x + \frac{2}{5}$
Answer
Solution:
Step 1:
Position $x$ on the left by rewriting the inequality: $x + \frac{2}{5} < 5.3$
Step 2:
Isolate $x$ by moving other terms to the opposite side.
Step 2.1:
Subtract $\frac{2}{5}$ from each side: $x < 5.3 - \frac{2}{5}$
Step 2.2:
Express $5.3$ as a fraction by multiplying by $\frac{5}{5}$: $x < 5.3 \cdot \frac{5}{5} - \frac{2}{5}$
Step 2.3:
Combine the fraction with $5.3$: $x < \frac{5.3 \cdot 5}{5} - \frac{2}{5}$
Step 2.4:
Merge the terms over a common denominator: $x < \frac{5.3 \cdot 5 - 2}{5}$
Step 2.5:
Simplify the fraction's numerator.
Step 2.5.1:
Calculate $5.3$ times $5$: $x < \frac{26.5 - 2}{5}$
Step 2.5.2:
Subtract $2$ from $26.5$: $x < \frac{24.5}{5}$
Step 2.6:
Divide $24.5$ by $5$: $x < 4.9$
Step 3:
Express the solution in various formats.
Inequality Notation: $x < 4.9$
Interval Notation: $(-\infty, 4.9)$
Knowledge Notes:
The problem involves solving a simple linear inequality. The steps taken to solve the inequality are based on the principles of algebraic manipulation, aiming to isolate the variable of interest (in this case, $x$) on one side of the inequality sign.
Inequality manipulation: Similar to equations, you can add, subtract, multiply, or divide both sides of an inequality by the same nonzero number without changing the direction of the inequality.
Common denominator: When dealing with fractions, it's often useful to express all terms with a common denominator to simplify the expression.
Simplifying expressions: Arithmetic operations such as multiplication and subtraction are used to simplify the expressions.
Solution representation: The solution to an inequality can be expressed in inequality form (e.g., $x < 4.9$) or interval notation (e.g., $(-\infty, 4.9)$), which denotes the range of values that satisfy the inequality.
Decimal to fraction conversion: In this problem, the decimal $5.3$ is converted to a fraction by multiplying by $\frac{5}{5}$, which is equivalent to $1$, to find a common denominator with $\frac{2}{5}$.
Interval notation: The interval notation $(-\infty, 4.9)$ indicates all real numbers less than $4.9$ without including $4.9$ itself. The parentheses signify that the endpoint is not part of the interval, as opposed to brackets, which would indicate inclusion.
