Problem 66 - Introductory Algebra for College Students

Question

Solve each equation. Use words or set notation to identify equations that have no solution, or equations that are true for all real numbers. $$5 x-5=3 x-7+2(x+1)$$

Step-by-Step Solution

Answer

The equation is true for all real numbers.

Step 1: Distribute

Firstly, distribute the 2 to both \(x\) and 1 in the term \(2(x+1)\) on the right side of the equation to eliminate the parentheses. The equation then becomes: \(5x - 5 = 3x - 7 + 2x + 2\).

Step 2: Gather like terms

Combine like terms on each side. On the right side of the equation, combine \(3x\) and \(2x\) to get \(5x\), then combine \(-7\) and \(2\) to get \(-5\). The equation simplifies to: \(5x - 5 = 5x - 5\).

Step 3: Subtract \(5x\) from both sides

To isolate the variable, subtract \(5x\) from both sides. This gives: \(-5 = -5\).

Step 4: Analyze the result

The equation \(-5 = -5\) is always true and it doesn't involve \(x\). Therefore, the original equation is true for all real numbers.

Key Concepts

Distributive Property

The distributive property is a key concept that helps simplify expressions and solve equations effectively. This property allows us to "distribute" a multiplier to each term inside a set of parentheses. For example, when we look at the expression \(2(x + 1)\), the 2 multiplies both \(x\) and 1 separately. This action results in:

• \(2 \times x = 2x\)
• \(2 \times 1 = 2\)

Putting it all together, \(2(x + 1)\) becomes \(2x + 2\). By applying the distributive property, we make it easier to combine like terms later on. Remember to always perform distribution if you see parentheses, as it is the first step in simplifying your expression.

Combining Like Terms

Once you've used the distributive property, the next step is to combine like terms. Like terms are terms that contain the same variable raised to the same power. They can be added or subtracted from each other because they belong to the same group. In the equation from our problem, after distribution, we have:

• \(5x - 5 = 3x - 7 + 2x + 2\)

Here, \(3x\) and \(2x\) are like terms because they both have the variable \(x\). When we add them together, we get \(5x\). Similarly, constants like \(-7\) and \(2\) can be combined to get \(-5\). By combining like terms, our equation simplifies to \(5x - 5 = 5x - 5\). This makes it clearer and easier to analyze or solve the equation.

Equations True for All Real Numbers

An equation is true for all real numbers when it simplifies to a true statement without any variables involved, like \(-5 = -5\). This means every possible value for the variable satisfies the equation. When you reach a point where subtraction or other operations eliminate the variable completely and leave you with identical terms on both sides, you've encountered an identity.Such equations indicate that no specific solution is required because every \(x\) works. Always look out for these scenarios, as they tell you the equation describes a fundamental truth rather than a specific condition that \(x\) must meet. Identifying these equations accurately can save you time and highlight their universal nature.