Question

What represents the range of \(x\) -values in this inequality? \(-3(x+4)>2 x\) (A) \(x<-\frac{12}{5}\) (B) \(x \leq-\frac{1}{3}\) (C) \(x>\frac{7}{8}\) (D) \(x \geq 3 \frac{1}{2}\)

Short answer

Option (A) \(x < -\frac{12}{5}\).

Step-by-step solution

Expand the inequality

Start by expanding the inequality \(-3(x + 4) > 2x\). Distribute the \(-3\) across \(x + 4\). This gives us \(-3x - 12 > 2x\).

Combine like terms

To simplify the inequality, move \(2x\) to the left side by subtracting \(2x\) from both sides: \(-3x - 2x - 12 > 0\). This simplifies to \(-5x - 12 > 0\).

Isolate the variable

Add 12 to both sides to isolate the \(x\)-term: \(-5x > 12\).

Solve for \(x\)

Divide both sides by \(-5\) and remember to flip the inequality sign (since dividing by a negative number reverses the inequality): \(x < -\frac{12}{5}\).

Choose the correct option

Compare the solution \(x < -\frac{12}{5}\) with the given options. The correct answer is option (A) \(x < -\frac{12}{5}\).

Key concepts

Inequalities

Understanding inequalities is crucial in solving various mathematical problems. An inequality is a mathematical statement that compares two expressions using inequality symbols such as ">", "<", "≥", or "≤". In simple terms, these symbols tell us how one value relates to another.

In the context of the problem, we're given the inequality \(-3(x+4) > 2x\). This means that the expression \(-3(x+4)\) is greater than \(2x\). Our goal is to find the values of \(x\) that make this statement true. Handling inequalities requires attention to detail, especially when negative numbers or multiplication and division are involved, as these can affect the direction of the inequality symbol.

When solving, always remember:

• If you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign.
• Simplify expressions by combining like terms, just as you would in equations.
• Check your solutions against any constraints provided in the problem statement.

With these tips in mind, solving inequalities becomes a systematic task of simplifying and logically stepping towards the solution.

Algebra

Algebra is the foundation of solving equations and inequalities. It involves manipulating mathematical expressions and formulas to isolate variables and find solutions. In this exercise, algebra is used to simplify and solve the given inequality to determine the range of \(x\).

The process starts by using the distributive property to expand \(-3(x + 4)\), resulting in \(-3x - 12\). This is a key algebraic skill: applying the distributive property helps break down more complex expressions into simpler ones.

Next, we move terms to one side of the inequality to combine like terms, such as moving \(2x\) to the left. This involves the inverse operation, which is a central algebraic concept; subtraction here helps us isolate terms involving \(x\). As steps progress, algebra helps us systematically transform and simplify the inequality, making it ready for the final solution steps.

Remember during such manipulations:

• Maintain balance by performing the same operation on both sides of the equation or inequality.
• Carefully track and simplify all terms to avoid common errors.
• Isolate the variable by methodically moving terms until you achieve a clear inequality or equation form that directly shows the solution.

Problem Solving

Problem solving in mathematics involves clear steps to achieve a solution, starting with understanding the problem statement and ending with verifying your solution. This systematic approach ensures that the path to solving an inequality is logical and reliable.

Begin by carefully interpreting the problem. Here, we were asked to determine the range of \(x\) that satisfies the given inequality \(-3(x+4) > 2x\). Recognizing what each part of the expression represents can help you develop a strategy.

In problem solving, it's useful to adhere to these principles:

• Break down the problem into manageable parts, solving step by step.
• Check each step as you progress, ensuring accuracy in your calculations.
• Verify if your solution aligns with the possible options provided if it's a multiple-choice question.

Once the inequality was solved in the given problem, the solution \(x < -\frac{12}{5}\) needed to be compared against multiple-choice answers. This final step in problem solving ensures that your understanding is applied correctly, allowing you to select the appropriate answer confidently. Proper problem-solving skills build accuracy and efficiency in tackling a broad spectrum of mathematical problems.