Question

place an x in the appropriate cell to indicate whether each equation has one solution, no solution or infinitely many solutions.
4(2x + 1)=5(x + 2)+3
5(x - 6)=2(6 + 3x)
4(2x - 3)=2(4x - 6)
one solution
no solutions
infinitely many solutions

Explanation:

Step1: Simplify \(4(2x + 1)=5x + 2+3\)

Expand the left - hand side: \(8x+4 = 5x + 5\).

Step2: Solve for \(x\)

Subtract \(5x\) from both sides: \(8x - 5x+4=5x - 5x + 5\), \(3x+4 = 5\). Then subtract 4 from both sides: \(3x=1\), \(x=\frac{1}{3}\), so one solution.

Step3: Simplify \(5(x - 6)=2(6 + 3x)\)

Expand both sides: \(5x-30 = 12 + 6x\).

Step4: Solve for \(x\)

Subtract \(5x\) from both sides: \(5x-5x - 30=12 + 6x-5x\), \(-30=x + 12\). Then subtract 12 from both sides: \(x=-42\), so one solution.

Step5: Simplify \(4(2x - 3)=2(4x - 6)\)

Expand both sides: \(8x-12 = 8x-12\).

Step6: Analyze the equation

Subtract \(8x\) from both sides: \(8x-8x-12=8x-8x - 12\), \(- 12=-12\). This is always true, so infinitely many solutions.

Answer:

1. For \(4(2x + 1)=5x + 2+3\):

• One solution

1. For \(5(x - 6)=2(6 + 3x)\):

• One solution

1. For \(4(2x - 3)=2(4x - 6)\):

• Infinitely many solutions