Summer Assignment Review Graphing Compound Inequalities And Absolute Value Inequalities

Compound Inequalities AND compound inequalities should create a range of values. −2≤𝑥≤3 𝑥≥−2 𝐴𝑁𝐷 𝑥≤3 Don’t assume that the given inequality creates this range. −2≥𝑥≥3 𝑥≤−2 𝐴𝑁𝐷 𝑥≥3 𝑥 falls between -2 and 3 on the number line. 𝑥 can not be both to the left of -2 and to the right of 3 on the number line . There is no solution.

Compound Inequalities OR compound inequalities should create two ranges of values. 𝑥≤−2 𝑂𝑅 𝑥≥3 AND compound inequalities should create a range of values. 𝑥≥−2 𝑂𝑅 𝑥≤3 𝑥 is either to the left of -2 or to the right of 3 on the number line. 𝑥 can be anywhere on the number line. The solution is All Real Numbers.

Practice Graph each compound inequality. 1) −3<𝑥<4 2) 3) 4) 5) −3<𝑥<4 All Real Numbers 𝑥 ≥−3 𝑜𝑟 𝑥<3 𝑥<−4 𝑜𝑟 𝑥 ≥2 𝑥≤0 𝑎𝑛𝑑 𝑥≥3 No Solutions 𝑥>−1 𝑎𝑛𝑑 𝑥 ≤5

Absolute Value Inequalities When we say that 𝑥 =3, we are saying that the distance between x and 0 is 3. When we say that 𝑥 <3, we are saying that the distance between x and 0 is less than 3. When we say that 𝑥 >3, we are saying that the distance between x and 0 is greater than 3. The inequalities < and ≤ lead to “and” relationships −3<𝑥<3 The inequalities > and ≥ lead to “or” relationships 𝑥<−3 𝑂𝑅 𝑥>3

Solving Absolute Value Inequalities Case 2 - modified method: Drop the absolute value bars, flip the inequality symbol and change the sign of the term on the right. 2𝑥−1<−7 2𝑥<−6 𝑥<−3

Solve and Graph an Abs. Val. Ineq. Example : Solve 𝑥−5 ≥7 . Graph your solution. Step 1: Is it and or or? For the second, flip the inequality and change the sign of the 7 Step 2: Solve both Step 3: Graph This is ≥ so it is “or”. Set up two inequalities. 𝑥−5 ≥7 𝑥−5≥7 𝑥−5≤−7 𝑥≥12 𝑥≤−2 𝑥≥12 or 𝑥≤−2 -3 -2 -1 · · · 11 12 13

Solve and Graph an Abs. Val. Ineq. Example : Solve . Graph your solution. Step 1: Is it and or or? Step 2: Solve +5 +5 +5 -4 -4 -4 Step 3: Graph −4𝑥−5 +3<9 We don’t know yet, get the abs. val. alone. Now : It is < , so it is an “and” inequality. Drop the abs. val. Bars and put -6 < on the left. −4𝑥−5 +3<9 −4𝑥−5 <6 −6<−4𝑥−5<6 −1<−4𝑥<11 0.25>𝑥>−2.75 −2.75<𝑥<0.25

Practice −5<𝑤<6 −0.2≤𝑚≤2.6 𝑥>5 𝑜𝑟 𝑥<−11 𝑥>5 𝑜𝑟 𝑥<−11 −5<𝑤<6 −0.2≤𝑚≤2.6 -12 -11 -10 · · · 4 5 6 -6 -5 -4 · · · 5 6 7