Question

3 - \frac{2}{5}x - \frac{12}{5} = \frac{10 - 2x}{5}
stretch optional
consider the equation 2x - 5(x - 1) = 50.
1 solve the equation for x.
2 chen was asked to solve the inequality: 2x - 5(x - substitute in any value for x less than -15 to determine the correct solution.

Explanation:

Sub - question 1: Solve the equation \(2x - 5(x - 1)=50\)

Step 1: Expand the left - hand side

We use the distributive property \(a(b - c)=ab - ac\). For \( - 5(x - 1)\), we have \(-5\times x-(-5)\times1=-5x + 5\). So the equation becomes \(2x-5x + 5 = 50\).

Step 2: Combine like terms

Combine the \(x\) terms on the left - hand side: \(2x-5x=-3x\). The equation is now \(-3x + 5 = 50\).

Step 3: Isolate the variable term

Subtract 5 from both sides of the equation. According to the subtraction property of equality, if \(a=b\), then \(a - c=b - c\). So \(-3x+5 - 5=50 - 5\), which simplifies to \(-3x=45\).

Step 4: Solve for \(x\)

Divide both sides of the equation by \(-3\). According to the division property of equality, if \(a=b\), then \(\frac{a}{c}=\frac{b}{c}(c
eq0)\). So \(\frac{-3x}{-3}=\frac{45}{-3}\), which gives \(x=- 15\).

Step 1: Simplify the left - hand side

First, combine the constant terms on the left - hand side. \(3=\frac{15}{5}\), so \(3-\frac{12}{5}=\frac{15}{5}-\frac{12}{5}=\frac{15 - 12}{5}=\frac{3}{5}\). The equation becomes \(\frac{3}{5}-\frac{2}{5}x=\frac{10 - 2x}{5}\).

Step 2: Eliminate the denominator

Multiply both sides of the equation by 5 to get rid of the denominators. According to the multiplication property of equality, if \(a = b\), then \(a\times c=b\times c\). So \(5\times(\frac{3}{5}-\frac{2}{5}x)=5\times\frac{10 - 2x}{5}\). This simplifies to \(3-2x = 10 - 2x\).

Step 3: Analyze the equation

Add \(2x\) to both sides of the equation. We get \(3-2x + 2x=10 - 2x+2x\), which simplifies to \(3 = 10\). Since \(3
eq10\), this equation has no solution.

Answer:

\(x = - 15\)

Sub - question 2: Solve the equation \(3-\frac{2}{5}x-\frac{12}{5}=\frac{10 - 2x}{5}\)