Solve the Inequality for x 5.3> x+2/5

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: $(-\mathrm{\infty },4.9)$

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.

1. 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.

2. Common denominator: When dealing with fractions, it's often useful to express all terms with a common denominator to simplify the expression.

3. Simplifying expressions: Arithmetic operations such as multiplication and subtraction are used to simplify the expressions.

4. Solution representation: The solution to an inequality can be expressed in inequality form (e.g., $x<4.9$) or interval notation (e.g., $(-\mathrm{\infty },4.9)$), which denotes the range of values that satisfy the inequality.

5. 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 $\frac{2}{5}$.

6. Interval notation: The interval notation $(-\mathrm{\infty },4.9)$ 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.