1.01 Rearranging Linear Inequalities Unit: Optimization

To isolate the y-variable Function Form: 𝒚= 𝒂𝒙+𝒃 Goal To isolate the y-variable Function Form: 𝒚= 𝒂𝒙+𝒃

How To: Goal is to get the y term on one side of the inequality, and everything else on the other side. To do this: Use algebra to bring the term with the y-variable to the left side of the inequality (if it isn’t already there), the same way you would if there was an = Bring all other terms to the right side of the inequality, the same way you would if there was an = Divide everything by the coefficient of y If the coefficient is negative, the inequality symbol FLIPS (ex: from < to >).

2𝑥+3𝑦−9<0 3𝑦<−2𝑥+9 3𝑦 3 < −2𝑥+9 3 𝑦<− 2 3 𝑥+3 Bring the term with the y-variable to the left side of equation Bring any other terms to the right side of the equation Divide everything by the coefficient of y (If the coefficient is negative, the inequality symbol FLIPS) 3𝑦<−2𝑥+9 3𝑦 3 < −2𝑥+9 3 𝑦<− 2 3 𝑥+3

5𝑥−2𝑦+13≤0 −2𝑦≤−5𝑥−13 −2𝑦 −2 ≤ −5𝑥−13 −2 𝑦≥ 5 2 𝑥+ 13 2 Bring the term with the y-variable to the left side of equation Bring any other terms to the right side of the equation Divide everything by the coefficient of y (If the coefficient is negative, the inequality symbol FLIPS) −2𝑦≤−5𝑥−13 −2𝑦 −2 ≤ −5𝑥−13 −2 𝑦≥ 5 2 𝑥+ 13 2

7−2𝑥>3𝑦 7−2𝑥−3𝑦>0 −3𝑦>2𝑥−7 −3𝑦 −3 > 2𝑥−7 −3 Bring the term with the y-variable to the left side of equation Bring any other terms to the right side of the equation Divide everything by the coefficient of y (If the coefficient is negative, the inequality symbol FLIPS) 7−2𝑥−3𝑦>0 −3𝑦>2𝑥−7 −3𝑦 −3 > 2𝑥−7 −3 𝑦<− 2 3 𝑥+ 7 3

3𝑥 2 − 4𝑦 5 >1 Bring the term with the y-variable to the left side of equation Bring any other terms to the right side of the equation Divide everything by the coefficient of y (If the coefficient is negative, the inequality symbol FLIPS) − 4𝑦 5 >− 3𝑥 2 +1 − 4𝑦 5 − 4 5 > − 3𝑥 2 +1 − 4 5 𝑦< 15 8 𝑥− 5 4