Problem 26 - Precalculus

Question

Solve the equation or inequality. $$5-(4-2 x)^{\frac{2}{3}}=1$$

Step-by-Step Solution

Answer

The solution is \(x = -2\).

Step 1: Isolate the Exponential Term

Start by isolating the term that has the exponent. In the equation \(5 - (4 - 2x)^{\frac{2}{3}} = 1\), subtract 5 from both sides to get \[-(4-2x)^{\frac{2}{3}} = 1 - 5\].This simplifies to \[-(4-2x)^{\frac{2}{3}} = -4\].

Step 2: Simplify Negative Signs

Multiply both sides of the equation by -1 to eliminate the negative sign on the left side:\[(4-2x)^{\frac{2}{3}} = 4\].

Step 3: Eliminate the Fractional Exponent

Raise both sides of the equation to the power of \(\frac{3}{2}\) to remove the fractional exponent. This is done as follows:\[(4-2x) = 4^{\frac{3}{2}}\].

Step 4: Simplify Exponentiation

Calculate \(4^{\frac{3}{2}}\). Write it as \((4^{\frac{1}{2}})^3\) which is \[2^3 = 8\].This gives \[4 - 2x = 8\].

Step 5: Solve for x

Subtract 4 from both sides to isolate terms involving \(x\):\[-2x = 8 - 4\], which simplifies to \[-2x = 4\].Now divide both sides by -2 to solve for \(x\):\[x = \frac{4}{-2}\], which simplifies to \[x = -2\].

Key Concepts

Fractional Exponents

Fractional exponents can be thought of as a combination of root and power. For example, a fractional exponent like \(a^{\frac{m}{n}}\) means the \(n\)-th root of \(a\) raised to the \(m\)-th power. This can also be understood in steps:

• The numerator in the fraction (\(m\)) corresponds to a power.
• The denominator (\(n\)) corresponds to a root.

For the equation \((4 - 2x)^{\frac{2}{3}}\), it means you take the cube root of \((4 - 2x)\) and then square the result. This understanding helps when you need to manipulate these exponents to solve equations.

Isolation of Terms

Isolation of terms is crucial in solving equations, as it allows us to focus on the unknown variable. The idea is to rearrange the equation so that the term involving the variable is alone on one side of the equation.

• Start by moving other terms to the opposite side.
• Perform inverse operations like addition/subtraction or multiplication/division.

In our given exercise: initially, isolate the term \(-(4 - 2x)^{\frac{2}{3}}\) by moving constant terms to the other side: \(-(4 - 2x)^{\frac{2}{3}} = -4\). Once isolated, it is simpler to handle in subsequent steps.

Solving Equations

Solving equations involves several systematic steps to find the value of the unknown variable. The primary goal is to perform operations that simplify and eventually isolate the variable:

• After isolating terms, simplify if needed by handling exponents or roots.
• In our example, removing the fractional exponent \(\frac{2}{3}\) by raising both sides to the reciprocal power (\(\frac{3}{2}\)). This gives \((4-2x) = 4^{\frac{3}{2}} = 8\).
• Continue simplifying and finally perform basic arithmetic operations.

Ensure every operation is justified by maintaining equality. Once the terms are simplified, the equation can easily be solved for \(x\).

Negative Signs in Equations

Negative signs can appear anywhere in an equation, yet they often become a source of confusion. It's essential to clarify their role and to handle them correctly to prevent sign errors.

• In the problem, \(-(4-2x)^{\frac{2}{3}}\) had a negative sign. Multiplying both sides by \(-1\) removes it.
• Keep track of negative signs when dividing or multiplying by negative values. For example, \(-2x = 4\) became \(x = -2\) after dividing by \(-2\).

Handling negative signs requires careful attention but with practice, managing them becomes more intuitive.