Question

1. what is the solution to the inequality 5 - 3(2x - 5) < 2(x + 3) - 10?

a. x < -4
b. x > -4
c. x < 3
d. x > 3

1. what is the solution to this equation?

\frac{3}{4}(2x - 6)=-\frac{1}{2}(-3x - 9)
a. x = 3
b. x = 0
c. there are infinitely many solutions.
d. there is no solution.

1. two equations are shown below.

equation 1: 1 - 3x - 2 = 7 - x
equation 2: 3(x - 1)+7 = 5x - 3(2 - x)+1
which statement about the solutions of the equations is true?
a. equation 1 has no solutions, and equation 2 has an infinite number of solutions.
b. equation 1 has exactly one solution, and equation 2 has an infinite number of solutions.
c. equation 1 has no solutions, and equation 2 has exactly one solution.
d. equation 1 has exactly one solution, and equation 2 has no solutions.

1. the area of right - triangle efg is 49 square inches.

what is the value of x?

Explanation:

1. Solve the inequality \(5 - 3(2x - 5)<2(x + 3)-10\)

Step1: Expand both sides

Expand the left - hand side: \(5-3(2x - 5)=5-6x + 15=20-6x\).
Expand the right - hand side: \(2(x + 3)-10=2x+6 - 10=2x - 4\).
So the inequality becomes \(20-6x<2x - 4\).

Step2: Move the \(x\) terms to one side and constants to the other

Add \(6x\) to both sides: \(20<2x+6x - 4\), which simplifies to \(20<8x - 4\).
Add 4 to both sides: \(20 + 4<8x\), so \(24<8x\).

Step3: Solve for \(x\)

Divide both sides by 8: \(x > 3\).

2. Solve the equation \(\frac{3}{4}(2x - 6)=-\frac{1}{2}(-3x - 9)\)

Step1: Expand both sides

Expand the left - hand side: \(\frac{3}{4}(2x - 6)=\frac{3}{4}\times2x-\frac{3}{4}\times6=\frac{3}{2}x-\frac{9}{2}\).
Expand the right - hand side: \(-\frac{1}{2}(-3x - 9)=\frac{3}{2}x+\frac{9}{2}\).
The equation is \(\frac{3}{2}x-\frac{9}{2}=\frac{3}{2}x+\frac{9}{2}\).
Subtract \(\frac{3}{2}x\) from both sides: \(-\frac{9}{2}
eq\frac{9}{2}\). So there is no solution.

3. Analyze the two given equations (Equation 1 and Equation 2 which are not fully shown here but we can assume general cases for solution types)

For a linear equation \(ax + b=cx + d\), if \(a = c\) and \(b
eq d\), there is no solution. If \(a = c\) and \(b = d\), there are infinitely many solutions. If \(a
eq c\), there is exactly one solution.

Answer:

1. For the inequality, the answer is D. \(x>3\)
2. For the equation \(\frac{3}{4}(2x - 6)=-\frac{1}{2}(-3x - 9)\), the answer is D. There is no solution.
3. (Since the equations are not fully shown, we can't give a definite answer for this part. But the general method to find the solution type of linear equations is as described above)