Video Transcript
Find algebraically the solution set of the equation 11𝑥 plus 44 over the absolute value of 𝑥 plus four is equal to 𝑥.
We begin by recalling that for any valid solution the denominator of our fraction cannot equal zero. This means that the absolute value of 𝑥 plus four is not equal to zero. It follows that 𝑥 plus four cannot be equal to zero. This means that 𝑥 could not be equal to negative four. It is important to keep this in mind when solving this question.
We will begin solving the equation by firstly rearranging it so that the absolute value function is on one side of the equation. Multiplying both sides of the equation by the absolute value of 𝑥 plus four and dividing both sides by 𝑥, we have the absolute value of 𝑥 plus four is equal to 11𝑥 plus 44 divided by 𝑥. Next, we need to consider what we mean by the absolute value function.
The absolute value is the distance from zero and is always positive. We therefore need to consider two situations: when 𝑥 plus four is greater than or equal to zero and when 𝑥 plus four is less than zero. This will give us two equations we need to solve.
Firstly, when 𝑥 plus four is greater than or equal to zero, we need to solve the equation 𝑥 plus four is equal to 11𝑥 plus 44 divided by 𝑥. Multiplying both sides of this equation by 𝑥, we have 𝑥 multiplied by 𝑥 plus four is equal to 11𝑥 plus 44. Distributing the parentheses on the left-hand side gives us 𝑥 squared plus four 𝑥. Our next step is to subtract 11𝑥 and 44 from both sides of the equation. This gives us the quadratic equation 𝑥 squared minus seven 𝑥 minus 44 is equal to zero.
We can solve this quadratic equation by factoring. The left-hand side becomes 𝑥 minus 11 multiplied by 𝑥 plus four. This is because negative 11 and four have a product of negative 44 and a sum of negative seven. This leads us to two solutions: 𝑥 equals 11, or 𝑥 equals negative four.
We have already established that 𝑥 cannot be equal to negative four, so we can ignore this solution. We said that this equation is true when 𝑥 plus four is greater than or equal to zero and when 𝑥 equals 11, 𝑥 plus four equals 15, which is indeed greater than or equal to zero. Therefore, the solution 𝑥 equals 11 is valid.
Next, we need to consider the situation when 𝑥 plus four is less than zero. If 𝑥 plus four is negative, then when we take its absolute value, we are actually evaluating the negative of 𝑥 plus four to get that positive value. So to find the value of 𝑥 that satisfies this, we will need to solve the equation negative 𝑥 plus four is equal to 11𝑥 plus 44 divided by 𝑥.
As with our first situation, we begin by multiplying both sides by 𝑥, giving us negative 𝑥 multiplied by 𝑥 plus four is equal to 11𝑥 plus 44. We can then distribute the parentheses on the left-hand side, giving us negative 𝑥 squared minus four 𝑥. This is equal to 11𝑥 plus 44. Adding 𝑥 squared and four 𝑥 to both sides of this equation gives us zero is equal to 𝑥 squared plus 15𝑥 plus 44.
This is another quadratic equation that we can solve by factoring. 𝑥 squared plus 15𝑥 plus 44, in its factored form, is equal to 𝑥 plus four multiplied by 𝑥 plus 11. This gives us two possible solutions: 𝑥 equals negative four, or 𝑥 equals negative 11. Once again, we know that 𝑥 equals negative four is not a valid solution. We said that this equation is true when 𝑥 plus four is less than zero. So if 𝑥 equals negative 11, then 𝑥 plus four is negative 11 plus four, which is negative seven, which indeed is less than zero. We have therefore found a valid solution; 𝑥 equals negative 11.
We can therefore conclude that the solution set of the equation 11𝑥 plus 44 divided by the absolute value of 𝑥 plus four equals 𝑥 contains the values negative 11 and 11. We could check these solutions by substituting them back into the original equation.
