How to find out when an equation has no solution

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ACT Math › How to find out when an equation has no solution

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Solve the following equation for :

$2\left ( x - 4\right ) = \frac{4x + 5}{2}$

No solution

CORRECT

$-\frac{3}{2}$

Infinite solutions

$\frac{13}{2}$

Explanation

In order to solve for , we must get by itself on one side of the equation.

$2\left ( x - 4\right ) = \frac{4x + 5}{2}$

First, we can distribute the on the left side of the equal sign and the $\frac{1}{2}$ on the right side.

$2x - 8= 2x +\frac{ 5}{2}$

When we try to get by itself, the terms on each side of the equation cancel out, leaving us with:

$- 8= \frac{ 5}{2}$

We know this is an untrue statement, so there is no solution to this equation.

Nosol1

There is no solution

CORRECT

–3

–1/2

Explanation

Nosol2

$\frac{x+2}{3}=\frac{x}{3}$ Solve for .

No solutions.

CORRECT

Explanation

Cross multiplying leaves , which is not possible.

Solve:

No Solution

CORRECT

Infinitely Many Solutions

$\frac{15}{9}$

Explanation

First, distribute the to the terms inside the parentheses.

Add 6x to both sides.

This is false for any value of . Thus, there is no solution.

Given the following system, find the solution:

x = y – 2

2x – 2y = 2

(1, 1)

no solution

CORRECT

(0, 0)

(1, 2)

(0, 1)

Explanation

When 2 equations in a system have the same slopes, they will either have no solution or infinite solutions. Since the y-intercepts are not the same, there is no solution to this system.

Solve $\left | 3-4x\right |<0$ .

No solutions

CORRECT

$x<\frac{4}{3}$

$x>\frac{4}{3}$

$x<\frac{3}{4}$

$x>\frac{3}{4}$

Explanation

By definition, the absolute value of an expression can never be less than 0. Therefore, there are no solutions to the above expression.

Find the solution to the following equation if x = 3:

y = (4x2 - 2)/(9 - x2)

no possible solution

CORRECT

Explanation

Substituting 3 in for x, you will get 0 in the denominator of the fraction. It is not possible to have 0 be the denominator for a fraction so there is no possible solution to this equation.

$\small h\left ( x\right )=\frac{28}{x+4}$

$\small \textup{For which of the following values of},x,\textup{is the above function undefined?}$

$\small -4$

CORRECT

$\small 4$

$\small 0$

$\small 28$

None of the other answers

Explanation

A fraction is considered undefined when the denominator equals 0. Set the denominator equal to zero and solve for the variable.

$\small x+4=0$

$\small x=-4$

Undefined_denom3

I. x = 0

II. x = –1

III. x = 1

I only

CORRECT

II only

III only

II and III only

I, II, and III

Explanation

Undefined_denom2

Solve for :

No solution

CORRECT

Infinite Solutions

$-9.25\cdot 10^{-2}$

Explanation

Like other "solve for x" problems, to begin it, the goal is to get x by itself on one side of the equals sign. In this problem, before doing so, the imaginary -1 in front of (-27x+27) must be distributed.