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When we talk about binomial squaring, we dive into the idea of multiplying a binomial by itself. A binomial is an algebraic expression with two terms, like \(x+2\). Squaring this means multiplying it by \(x+2\) again.

To carry out this operation, we apply the formula: \((a + b)^2 = a^2 + 2ab + b^2\). For the given inequality \((x+2)^2\), this translates to:

• \(x^2 \) for the \(x\) term squared.
• \(2(x)(2)\) for the product of twice the first term and the second term.
• \(4\) for the square of the second term.

So, \((x + 2)^2 = x^2 + 4x + 4\). In the context of solving inequalities, particularly \((x+2)^{2} \leq 25\), squaring a binomial helps us work with a perfect square on one side, simplifying subsequent steps in solving it.

The real number line is a visual representation of all possible real numbers. This line extends infinitely in both positive and negative directions. For inequalities, it's a useful tool to show the set of solutions that satisfy a condition.

In the original problem \( -7 \leq x \leq 3\), the real number line helps us to visually interpret the range of solutions.

• Begin by identifying specific points like -7 and 3 on the line.
• These values signify boundaries of the solution.

Real number line helps indicate whether values are included (closed circles) or not (open circles). Since the given inequality \(-7 \leq x \leq 3\) includes the boundary values, we use closed circles at -7 and 3, depicting inclusion. This tool aids in comprehending which numbers satisfy the inequality condition we found.

Graphical representation enhances understanding by translating a problem solution into a visual format. After solving an inequality, exhibiting it graphically ties together mathematical concepts and improves interpretation.

To graphically represent the inequality \( -7 \leq x \leq 3\), follow these simple steps:

• Draw a straight horizontal line to symbolize the number line.
• Mark the endpoints, like -7 and 3.
• Use filled circles to indicate -7 and 3 are part of the solution set.
• Connect these points with a line segment to show all values between and including -7 and 3 satisfy the inequality.

This graphical method makes the solution clearer, showing all solutions in a line for easy identification. It's an effective way to ensure all possible solutions of the inequality are acknowledged and well-represented.