Problem 56 - A Graphical Approach to College Algebra

Question

Classify each equation as a contradiction, an identity, or a conditional equation. Give the solution set. Use a graph or table to support your answer. $$1.5(6 x-3)-7 x=3-(7-x)$$

Step-by-Step Solution

Answer

The equation is conditional with solution set \(x = 0.5\).

Step 1: Simplify Both Sides

Start by expanding and simplifying each side of the equation separately. On the left side of the equation: \[1.5(6x-3) - 7x = 9x - 4.5 - 7x = 2x - 4.5\].On the right side of the equation:\[3 - (7 - x) = 3 - 7 + x = x - 4\].So, the equation reduces to:\[2x - 4.5 = x - 4\].

Step 2: Isolate x

Now, isolate the variable \(x\). Subtract \(x\) from both sides of the equation:\[2x - 4.5 - x = x - 4 - x\], which simplifies to:\[x - 4.5 = -4\].

Step 3: Solve for x

Add 4.5 to both sides of the equation to solve for \(x\):\[x - 4.5 + 4.5 = -4 + 4.5\], which gives:\[x = 0.5\].

Step 4: Classify the Equation

Since we have found a specific value for \(x\), \(x = 0.5\), that satisfies the equation, it is a conditional equation. It holds true only under the condition that \(x = 0.5\).

Step 5: Graphical Confirmation

To confirm, consider plotting the simplified forms \(y = 2x - 4.5\) and \(y = x - 4\) on a graph. They intersect at the point \((0.5, -3.5)\), confirming that \(x = 0.5\) is the solution.

Key Concepts

Solution Set

A solution set in mathematics refers to the collection of all possible values that a variable can take, which satisfy a given equation. In our exercise, the equation simplifies to find the solution:

• Through simplification, using algebraic manipulations, we isolated the variable and solved for it.
• The solution we found is: \(x = 0.5\).

This solution set contains just one value, \(x = 0.5\), since this is the only number that, when substituted back into the original equation, makes it true. Therefore, it's an important part of verifying our work, as only this specific value satisfies the equation within the context given.

Equation Classification

Equations can be classified into three main types: contradiction, identity, and conditional. Understanding these classifications helps us determine the nature and possible solutions of an equation.

• A **contradiction** is an equation that has no solution; no value will satisfy the equation (e.g., \(x + 2 = x + 5\)).
• An **identity** is an equation that is true for all possible values of the variable, meaning every number satisfies it (e.g., \(2x + 3 = 2x + 3\)).
• A **conditional equation** is true only for specific values of the variable. In our case, the equation \(1.5(6x-3) - 7x = 3 - (7-x)\) simplifies to \(x = 0.5\), making it conditional.

Here, because there is only one value that satisfies the equation, it is classified as a conditional equation. This classification is crucial because it tells us that the equation is valid only under certain conditions, which is here specified by \(x = 0.5\).

Graphical Representation

By using graphs, we can visually interpret the solution to an equation and verify its classification. For our exercise, once simplified, the equation becomes two linear equations:

• \(y = 2x - 4.5\)
• \(y = x - 4\)

When we plot these two equations on a coordinate plane, we can observe that they intersect at exactly one point. The intersection point, \((0.5, -3.5)\), indicates the solution of the variable \(x\). This visualization confirms our algebraic solution.Graphing is not just a way to verify solutions but also offers insight by showing that the equations are linear and intersect at a point. This single intersection confirms that the solution set \(x = 0.5\) is both accurate and makes equation \(1.5(6x - 3) - 7x = 3 - (7-x)\) a conditional equation.