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Mathematical manipulation is a key skill required to solve inequalities effectively. It involves changing the form of an equation or inequality to make it easier to work with while maintaining its equality or inequality property. This often includes operations such as addition, subtraction, multiplication, or division. By performing the same operation on both sides of an inequality, you keep the relationship between variables consistent.
When solving inequalities, one important aspect is to be cautious when multiplying or dividing both sides by a negative number. This action will reverse the inequality symbol. For instance, when solving the inequality \(4.8x - 0.12y < 7.2\), after subtracting \(4.8x\) from both sides, dividing by \(-0.12\) reverses the less than sign to a greater than sign, giving \(y > 60 - 40x\).
Understanding these foundational techniques helps make algebraic processes smoother and more intuitive.

Rearranging equations is about changing the arrangement of terms in an equation or inequality to isolate a specific variable, making it easier to solve. This process often starts with identifying the term that contains the variable you want to solve for, in this case, \( y \), and then using arithmetic to move other terms away from it.
For example, in the inequality \(84x + 7y \geq 70\), isolating \(y\) required us to subtract \(84x\) from both sides. This step simplifies our inequality to \(7y \geq 70 - 84x\), allowing us to handle the equation piece-by-piece. After rearranging, solving becomes straightforward by focusing on operations like division or multiplication on the necessary terms.

• Identify the term with the variable of interest.
• Use inverse operations to remove other terms one by one.
• If needed, rearrange any complex expressions resulting from initial changes.

Rearranging is not just about moving terms, but organizing them logically to achieve clear solutions.

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators (such as plus or minus signs). When solving inequalities, you often work with algebraic expressions on each side of the inequality symbol.
An important skill is simplifying these expressions to make them easier to manage and understand. Consider the inequality \(7y \geq 70 - 84x\) turned into the expression \(y \geq 10 - 12x\) after dividing each part by \(7\). Each transformation ensures that expressions on either side are in their simplest form, leading to easier interpretation and resolution.
Managing algebraic expressions also requires combining like terms and sometimes factoring if a more concise expression is needed. Accurate manipulation leads to clear and complete solutions. Remember that each piece of the expression plays a role in preserving the inequality's validity throughout the solution process, so careful calculation is crucial at each step.