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Find algebraically the solution set of the equation −11𝑥 − 17|𝑥| = −196.

Find algebraically the solution set of the equation negative 11𝑥 minus 17 multiplied by the modulus of 𝑥 is equal to negative 196.

We will begin trying to solve this equation by trying to isolate the modulus or absolute value of 𝑥. Firstly, we add 196 and 17 multiplied by the modulus of 𝑥 to both sides. This gives us negative 11𝑥 plus 196 is equal to 17 multiplied by the modulus of 𝑥. We can then divide both sides of this equation by 17 such that the modulus of 𝑥 is equal to negative 11𝑥 plus 196 all divided by 17. We know that the modulus or absolute value of a number is its distance from zero. This means that the sign does not matter. So either 𝑥 is equal to negative 11𝑥 plus 196 over 17 or negative 𝑥 is equal to negative 11𝑥 plus 196 over 17.

We multiply both sides of these equations by 17 so that 17𝑥 and negative 17𝑥 are equal to negative 11𝑥 plus 196. Adding 11𝑥 to both sides of the left-hand equation gives us 28𝑥 is equal to 196. We can then divide both sides by 28 giving us a value of 𝑥 equal to seven. By adding 11𝑥 to both sides of the right-hand equation, we get negative six 𝑥 is equal to 196. We can then divide both sides of this equation by negative six so that 𝑥 is equal to negative 98 over three.

We now have two solutions that satisfy the equation negative 11𝑥 minus 17 multiplied by the modulus of 𝑥 is equal to negative 196. They are negative 98 over three and seven, which can be written using set notation.

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