Question

d) $2(x - 4) + x = 3(x - 2) - 2$
f) $6(x - 1) + 21 = 6x + 15$
: $2x - 5(x - 1) < 50$. she gave an answer of $x < -15$.
15 to determine whether chen is correct. if not,

Explanation:

Solving Equation (d): \( 2(x - 4) + x = 3(x - 2) - 2 \)

Step 1: Expand both sides

Expand the parentheses using the distributive property (\(a(b + c)=ab + ac\)):
Left side: \(2(x - 4)+x = 2x-8 + x\)
Right side: \(3(x - 2)-2=3x-6 - 2\)
So the equation becomes \(2x-8 + x=3x-6 - 2\)

Step 2: Combine like terms

Left side: \(2x+x-8 = 3x-8\)
Right side: \(3x-(6 + 2)=3x-8\)
Now the equation is \(3x-8=3x-8\)

Step 3: Analyze the equation

Subtract \(3x\) from both sides: \(3x-3x-8=3x-3x-8\)
Which simplifies to \(-8=-8\)
This is a true statement for all real numbers \(x\). So the solution is all real numbers.

Solving Equation (f): \( 6(x - 1)+21 = 6x + 15 \)

Step 1: Expand the left - hand side

Using the distributive property \(a(b + c)=ab+ac\), we have:
\(6(x - 1)+21=6x-6 + 21\)

Step 2: Combine like terms on the left - hand side

\(6x-6 + 21=6x+( - 6 + 21)=6x + 15\)
Now the equation is \(6x + 15=6x + 15\)

Step 3: Analyze the equation

Subtract \(6x\) from both sides: \(6x-6x + 15=6x-6x + 15\)
Which simplifies to \(15 = 15\)
This is a true statement for all real numbers \(x\). So the solution is all real numbers.

Solving the inequality \(2x-5(x - 1)<50\) (to check Chen's answer \(x < - 15\))

Step 1: Expand the left - hand side

Using the distributive property \(a(b - c)=ab-ac\), we get:
\(2x-5(x - 1)=2x-5x + 5\)

Step 2: Combine like terms

\(2x-5x + 5=-3x + 5\)
So the inequality becomes \(-3x + 5<50\)

Step 3: Isolate the variable term

Subtract 5 from both sides: \(-3x+5 - 5<50 - 5\)
Which gives \(-3x<45\)

Step 4: Solve for \(x\)

Divide both sides by \(-3\). Remember that when we divide or multiply an inequality by a negative number, the direction of the inequality sign changes.
\(\frac{-3x}{-3}>\frac{45}{-3}\)
So \(x>- 15\)
Chen's answer was \(x < - 15\), which is incorrect. The correct solution is \(x>-15\)

Answer:

s:

• For equation (d): All real numbers.
• For equation (f): All real numbers.
• For the inequality \(2x - 5(x - 1)<50\): Chen is incorrect. The correct solution is \(x>-15\)