Problem 92 - Algebra A Combined Function

Question

Solve. \(\frac{x}{5}-\frac{3}{10} \geq \frac{x}{2}-1\)

Step-by-Step Solution

Answer

The solution to the inequality is \(x \leq \frac{7}{3}\).

Step 1: Eliminate Fraction Denominators

To simplify the inequality, eliminate unnecessary fractions by finding a common denominator for all the fractions. Since the denominators are 5, 10, and 2, the least common multiple is 10. Multiply every term in the inequality by 10 to clear the fractions:\[10 \left( \frac{x}{5} \right) - 10 \left( \frac{3}{10} \right) \geq 10 \left( \frac{x}{2} \right) - 10(1)\]This simplifies to:\[2x - 3 \geq 5x - 10\]

Step 2: Rearrange to Isolate Variable

The goal is to get all terms involving \(x\) on one side of the inequality. Start by subtracting \(5x\) from both sides:\[2x - 5x - 3 \geq -10\]This simplifies to:\[-3x - 3 \geq -10\]

Step 3: Simplify Constant Terms

Next, isolate the \(x\)-term by adding 3 to both sides of the inequality:\[-3x - 3 + 3 \geq -10 + 3\]This simplifies to:\[-3x \geq -7\]

Step 4: Solve for the Variable

Finally, solve for \(x\) by dividing both sides of the inequality by \(-3\). Remember that dividing by a negative number reverses the inequality sign:\[x \leq \frac{7}{3}\]

Step 5: Check the Solution

It's always good to verify the solution. Substitute a value smaller than \(\frac{7}{3}\) into the original inequality to ensure it holds true. For example, if you substitute \(x = 2\), both sides should satisfy the inequality. Substitution shows:\[\frac{2}{5} - \frac{3}{10} = \frac{4}{10} - \frac{3}{10} = \frac{1}{10}\]\[\frac{2}{2} - 1 = 0\]Since \(\frac{1}{10} \geq 0\) is true, the solution \(x \leq \frac{7}{3}\) is verified.

Key Concepts

Algebra

Algebra is the part of mathematics that deals with symbols and the rules for manipulating these symbols. In our exercise, algebra helps us unify numbers and variables to solve for the unknowns.
\(x\) in this case represents a variable, which is an unknown quantity that we need to solve for.
Key components we used:

• Variables: Symbols representing numbers in equations, here being \(x\).
• Coefficients: Numbers in front of variables showing multiplication, like 2 in \(2x\).
• Constants: Numbers on their own, such as -3 or -10.

Algebra allows us to set up equations and manipulate them to find the value of variables.We apply these operations systematically to isolate the variable and find solutions.

Fraction Elimination

Fractions can complicate calculations, especially in inequalities. Eliminating fractions makes it easier to manipulate the equation.
To eliminate fractions:

• Identify the denominators in the inequality. Here, the denominators are 5, 10, and 2.
• Find the least common multiple (LCM) of these denominators. For this exercise, it's 10.
• Multiply every term of the inequality by the LCM to clear the fractions.

Using the LCM eliminates all denominators, transforming terms like \(\frac{x}{5}\) to \(2x\). This makes the inequality easier to handle.The simplified inequality becomes more manageable and devoid of fractions. This step is crucial before further operations like adding or subtracting terms.

Solving Inequalities

Solving inequalities involves determining the range of values that satisfy the inequality condition.
The inequalities have special rules, especially compared to equations:

• When you multiply or divide both sides by a negative number, the direction of the inequality sign changes.
• Think of the inequality symbol as a balance; whatever you do to one side, you must do to the other.

In the problem, starting with \(2x - 3 \geq 5x - 10\), we manipulated terms:
- We subtracted \(5x\) from both sides, leading to \(-3x - 3\) being compared with \(-10\).- We added 3, isolating terms, simplifying to \(-3x \geq -7\).
Concluding with the division by \(-3\), and flipping the inequality gives the solution \(x \leq \frac{7}{3}\).Through consistent steps, inequalities lead us to a comprehensive solution set.

Variable Isolation

Variable isolation is about making the variable stand on one side of the inequality. This step is crucial to find the solution for \(x\).
The process includes:

• Writing all terms involving \(x\) on one side. Here, we moved all terms involving \(x\) by subtracting \(5x\).
• Getting rid of constant terms on the same side as \(x\) by adding or subtracting these terms. For instance, we added 3 to remove \(-3\).
• Lastly, divide to make the coefficient of \(x\) one. In our example, we divided by \(-3\).

During this, take care to reverse the inequality when dividing by negative numbers.Each of these steps serves to hone in on the exact value or range that \(x\) can hold in the context of the inequality.Through isolation, we express \(x\) independently, thus determining the set of possible solutions for \(x\).