How to Tackle the Latest POTW Inequality Challenge?

  • MHB
  • Thread starter anemone
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In summary, the direction of the inequality symbol is determined by the sign of the coefficient of the variable. Multiplying or dividing both sides of an inequality by a negative number requires flipping the direction of the symbol. Solving an inequality with one variable involves finding all possible values that satisfy the inequality, while with multiple variables, the goal is to find the range of values for each variable. To graph an inequality on a number line, you must first solve it as an equation and then plot the boundary points. Checking if numbers satisfy an inequality can be done by plugging them in, but it is not a conclusive proof for all possible values.
  • #1
anemone
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MHB
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Here is this week's POTW:

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Solve the inequality $\sqrt{4x^2-8x+5}+\sqrt{3x^2+12x+16}\ge 6\sqrt{x}-x-6$.

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  • #2
No one answered last week's POTW. (Sadface)

However, I will give the community another week's time to take another stab at the problem. I am looking forward to receiving submissions from the members!
 
  • #3
No one answered last two week's POTW. (Sadface) However, you can find the suggested solution (from other) as follows:

The solution set consists of all non-negative real numbers, as we shall show in the following.

Note that we need $x\ge 0$ in order for the right hand side of the inequality to be defined. Moreover, for all non-negative real numbers $x$, we have

$\begin{align*}\sqrt{4x^2-8x+5}+\sqrt{3x^2+12x+16}&=\sqrt{4(x-1)^2+1}+\sqrt{3(x+2)^2+4}\\& \ge 1+2 \\&=3\end{align*}$

On the other hand, $6\sqrt{x}-x-6=-(\sqrt{x}-3)^2+3\le 3$. This completes the proof.
 

Related to How to Tackle the Latest POTW Inequality Challenge?

1. How do you determine the solution set for an inequality?

The solution set for an inequality can be determined by graphing the inequality on a number line and shading the region that satisfies the inequality. The solution set will be all the values within the shaded region.

2. What is the difference between solving an inequality algebraically and graphically?

Solving an inequality algebraically involves using mathematical operations to isolate the variable and determine the solution set. Graphing an inequality involves plotting points on a number line and determining the solution set by shading the appropriate region.

3. How do you know when to use <, >, ≤, or ≥ symbols when solving an inequality?

The < symbol is used for "less than", the > symbol is used for "greater than", the ≤ symbol is used for "less than or equal to", and the ≥ symbol is used for "greater than or equal to". These symbols are used to represent the relationships between two quantities in an inequality.

4. Can you solve an inequality with multiple variables?

Yes, an inequality with multiple variables can be solved by treating it like a system of equations. You can use algebraic methods to isolate one variable in terms of the other, and then solve for the remaining variable.

5. How do you check if a value is a solution to an inequality?

To check if a value is a solution to an inequality, you can substitute the value into the original inequality and see if it satisfies the inequality. If the value makes the inequality true, then it is a solution. If the value makes the inequality false, then it is not a solution.

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