Match the description of the one variable equation with the number of solutions it will have.
ax+c1=bx+c2
(3 points)
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The number of x's on both sides of the equation is different a≠b
The number of x's is the same on both sides of the equation, and the constant is different a=b, c1≠c2
The number of x's is the same on both sides of the equation and the constant is also the same a=b, c1=c2
The number of x's on both sides of the equation is different a≠b - 1 solution
The number of x's is the same on both sides of the equation, and the constant is different a=b, c1≠c2 - infinitely many solutions
The number of x's is the same on both sides of the equation and the constant is also the same a=b, c1=c2 - 1 solution
Step 1: Distributive Property to get rid of the parentheses
2x - 5 = 12x + 15
Step 2: Combine like terms by subtracting 12x from both sides
-10x - 5 = 15
Step 3: Add 5 to both sides to isolate the variable
-10x = 20
Step 4: Divide both sides by -10 to solve for x
x = -2
Solving the inequality for x, which statement shows a good first step and which is the correct solution?
x−6<x4
(2 points)
Responses
Divide both sides of the inequality by 4 to simplify: x4−32<x
Divide both sides of the inequality by 4 to simplify: x4−32<x
Multiply both sides by 4 to clear the inequality of fractions: 4x−24<x
Multiply both sides by 4 to clear the inequality of fractions: 4x−24<x
Divide both sides of the inequality by x to simplify: 1−6x<4
x<6
Divide both sides of the inequality by x to simplify: 1−6x<4
x<6
Multiply both sides by x to get the "x" terms on the same side of the inequality: x−6<4
Divide both sides of the inequality by 4 to simplify: x/4 - 3/2 < x
This is the correct first step.
The correct solution is: x < 12/5 or x < 2.4
The number of x's on both sides of the equation is different a≠b - 1 solution
The number of x's is the same on both sides of the equation, and the constant is different a=b, c1≠c2 - No solution
The number of x's is the same on both sides of the equation and the constant is also the same a=b, c1=c2 - Infinite solutions
To determine the number of solutions to the one-variable equation, let's analyze the given options:
1. The number of x's on both sides of the equation is different (a ≠ b): In this case, there will be only one solution for the equation. The values of the constants (c1, c2) do not affect the number of solutions.
2. The number of x's is the same on both sides of the equation, and the constant is different (a = b, c1 ≠ c2): In this case, there will be no solution for the equation. The differing constants (c1, c2) make it impossible for the equation to be satisfied.
3. The number of x's is the same on both sides of the equation, and the constant is also the same (a = b, c1 = c2): In this case, there will be infinite solutions for the equation. Any value of x will satisfy the equation since both sides are identical.
Based on these descriptions, we can match the number of solutions to the corresponding equation:
- Option 1: One solution
- Option 2: No solution
- Option 3: Infinite solutions
2x−5=3(4x+5)
Step 1:
Property to get
Step 2:
Property to get
Step 3:
Property to get
Step 4:
Property to get