Why Are There No Real Solutions to the Equation \( |x^2 + 4x| = -12 \)?

In summary, an absolute value equation is an equation that involves the absolute value of a variable, represented as a positive value. To solve it, the absolute value must be isolated and written as a positive and negative equation. Real-life applications include finding distances and determining ranges of values. If there is no solution, it means the equation has no real solutions. Absolute value equations can only have two solutions, as the absolute value of a number can only have two values.
  • #1
mathdad
1,283
1
Explain why there are no real numbers that satisfy the equation
$$|x^2 + 4x| = - 12$$
How is this done algebraically?
 
Last edited by a moderator:
Mathematics news on Phys.org
  • #2
RTCNTC said:
Explain why there are no real numbers that satisfy the equation

|x^2 + 4x| = - 12

because ...

$|\text{whatever}| \ge 0$

How is this done algebraically?

solve each case ...

case 1 ... $x^2+4x \ge 0 \implies x^2+4x+12 = 0$ ... recommend looking at the discriminant

case 2 ... $x^2+4x < 0 \implies -x^2-4x+12 = 0$ ... check the two real solutions in the original abs equation.
 
  • #3
See attachment for my little outline.

View attachment 7492
 

Attachments

  • MathMagic171109_1.png
    MathMagic171109_1.png
    10.2 KB · Views: 104
Last edited by a moderator:
  • #4
$|whatever| \ge 0 \implies |whatever| \nless 0$

... that's it.
 
  • #5
Great. We now move on to the next topic.
 

FAQ: Why Are There No Real Solutions to the Equation \( |x^2 + 4x| = -12 \)?

What is an absolute value equation?

An absolute value equation is an equation that involves the absolute value of a variable. The absolute value of a number is its distance from 0 on a number line, and it is always represented as a positive value. In an absolute value equation, the variable is within the absolute value bars, and the equation will have two solutions.

How do you solve an absolute value equation?

To solve an absolute value equation, you must isolate the absolute value on one side of the equation. Then, rewrite the absolute value as a positive and negative equation and solve for both cases. The solutions are the values that make the absolute value expression equal to the given number.

What are some real-life applications of absolute value equations?

Absolute value equations can be used to solve problems involving distance, such as finding the minimum distance between two points or determining the absolute value of a number in real-world scenarios. They are also useful in determining the range of values for a given situation.

What happens when there is no solution to an absolute value equation?

If there is no solution to an absolute value equation, it means that the equation has no real solutions. This can occur when the absolute value expression is always positive and cannot equal the given number. In other words, the absolute value equation has no solutions if the number inside the absolute value bars is greater than the number on the other side of the equation.

Can absolute value equations have more than two solutions?

No, absolute value equations can only have two solutions. This is because the absolute value of a number can only have two values - the positive and negative version of that number. So, when solving an absolute value equation, there will only be two possible solutions.

Similar threads

MHB Why can't real numbers satisfy this absolute value equation?
Replies
3
Views
1K
MHB Is it Possible for an Absolute Value Equation to Equal a Negative Number?
Replies
4
Views
2K
MHB Absolute Value Equation |3x - 2|/|2x - 3| = 2
Replies
10
Views
2K
How to solve absolute value equation with two absolute values
Replies
6
Views
5K
MHB Find a,b,c,d for Max $a+b-c+d$
Replies
1
Views
877
MHB Inequality of positive real numbers
Replies
1
Views
972
MHB Solve Absolute Value Equation |x^2 - 2x| = |x^2 + 6x|
Replies
6
Views
2K
MHB Find Integer Values for 1999 Distinct Real Solutions
Replies
1
Views
982
MHB Absolute value of real numbers
Replies
1
Views
1K
MHB Solving Inequality 4x-12≤6x+20
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top