Section 1.7: Inequalities I will not cover how to graph inequalities on the real number line. Please see Section 1.7 of the Ron Larson textbook to review this. The focus of this presentation is how to solve linear inequalities and absolute value inequalities for x.

Solve the following inequalities 𝑥−5≥7 Add 5 to both sides: 𝑥≥12 𝑥+7≤12 Subtract 7 from both sides: 𝑥≤5

Solve the following inequalities 4𝑥<12 Divide both sides by 4: 𝑥<3 10𝑥>−40 Divide both sides by 10: 𝑥>−4 −2𝑥>−3 Divide both sides by -2. Any time you divide or multiply by a negative number, it will change the direction of the inequality sign: 𝑥< 3 2 −6𝑥<15 Divide both sides by -6: 𝑥> 15 −6 or 𝑥>− 5 2

Solve the following inequalities 2𝑥−5>7 Add 5 to both sides: 2𝑥>12 Divide by 2: 𝑥>6 2. −2𝑥+3≤−15 Subtract 3 from both sides: −2𝑥≤−18 Divide by -2: 𝑥≥9 REMEMBER: Dividing by a negative number changes the direction of the inequality sign.

Solve the following inequalities 2𝑥+7<3+4𝑥 Subtract 4x from both sides: −2𝑥+7<3 Subtract 7 from both sides: −2𝑥<−4 Divide by -2: 𝑥>2 2𝑥−1≥1−5𝑥 Add 5x to both sides: 7𝑥−1≥1 Add 1 to both sides: 7𝑥≥2 Divide by 7: 𝑥≥ 2 7

Solve the following inequalities 4−2𝑥<3 3−𝑥 Distribute first: 4−2𝑥<9−3𝑥 Add 3x to both sides: 4+𝑥<9 Subtract 4: 𝑥<5 2. −3 2−4𝑥 ≥−5(𝑥−1) Distribute first: −6+12𝑥≥−5𝑥+5 Add 5x to both sides: −6+17𝑥≥5 Add 6 to both sides: 17𝑥≥11 Divide by 17: 𝑥≥ 11 17

Solve the following inequality 3 4 𝑥−6<𝑥−7 Subtract 1𝑥 ( or 4 4 𝑥) from both sides: − 1 4 𝑥−6<−7 Add 6 to both sides: − 1 4 𝑥<−1 Multiply both sides by -4, the reciprocal of − 1 4 : −4 − 1 4 𝑥 <−4(−1) 𝑥>4 REMEMBER: The inequality sign switches when you multiply by a negative number.

Solve the following inequality 1 2 8𝑥+1 ≥3𝑥+ 5 2 This time, let’s take a different approach. Let’s multiply both sides by the LCD = 2. 2∙ 1 2 8𝑥+1 ≥2 3𝑥+ 5 2 8𝑥+1 ≥6𝑥+5 2𝑥≥4 𝑥≥2

Solve the following combined inequalities −3<𝑥+4<7 Subtract 4 from the left, the middle, and the right: −7<𝑥<3 1<2𝑥+3<9 Subtract 3 from the left, the middle, and the right: −2<2𝑥<6 Divide the left, middle, and right by 2: −1<𝑥<3

Solve the following combined inequality −8≤− 3𝑥+5 <13 Distribute first: −8≤−3𝑥−5<13 Add 5 to the left, the middle, and the right: −3≤−3𝑥<18 Divide the left, middle, and right by -3: 1≥𝑥>−6 The inequality signs switched because you divided by a negative number. Final Answer: −6<𝑥≤1

Solve the following combined inequality 0<2−3 𝑥+1 ≤20 Distribute first: 0<2−3𝑥−3≤20 0<−3𝑥−1≤20 Add 1 to the left, the middle, and the right: 1<−3𝑥≤21 Divide the left, middle, and right by -3: 1 −3 >𝑥≥−7 The inequality signs switched because you divided by a negative number. Final Answer: −7≤𝑥<− 1 3

Solve the following combined inequality −4< 2𝑥−3 4 <4 Multiply the left, middle, and right by 4: −16<2𝑥−3<16 Add 3 to the left, middle, and right: −13<2𝑥<19 Divide the left, middle, and right by 2: − 13 2 <𝑥< 19 2

Solve the following combined inequality −1<2− 𝑥 3 <1 Subtract 2 from the left, middle, and right: −3<− 𝑥 3 <−1 Multiply the left, middle, and the right by -3: −3 −3 <−3 − 𝑥 3 <−3 −1 9>𝑥>3 Final answer: 3<𝑥<9

Absolute Value Inequalities 1. If 𝑥 <𝑘, then 𝑥<𝑘 and 𝑥>−𝑘. In other words, −𝑘<𝑥<𝑘 Example: 𝑥 <4 → 𝑥<4 and 𝑥>−4 → −4<𝑥<4 2. If 𝑥 >𝑘, then 𝑥>𝑘 or 𝑥<−𝑘. Example: 𝑥 >5 → 𝑥>5 or 𝑥<−5

Solve the following Absolute Value inequalities with a < or ≤ sign 1. 𝑥−10 ≤4 → 𝑥−10≤4 and 𝑥−10≥−4 𝑥≤14 and 𝑥≥6 6≤𝑥≤14 2. 1−2𝑥 <5 → 1−2𝑥<5 and 1−2𝑥>−5 −2𝑥<4 and −2𝑥>−6 𝑥>−2 and 𝑥<3 −2<𝑥<3

Solve the Absolute Value inequality 1− 2𝑥 3 <1 1− 2𝑥 3 <1 and 1− 2𝑥 3 >−1 − 2𝑥 3 <0 and − 2𝑥 3 >−2 − 3 2 − 2𝑥 3 <− 3 2 0 and − 3 2 − 2𝑥 3 >− 3 2 −2 𝑥>0 and 𝑥<3 0<𝑥<3

Solve the Absolute Value inequality 2𝑥−3 −5<−4 Get the absolute value expression by itself: 2𝑥−3 <1 2𝑥−3<1 and 2𝑥−3>−1 2𝑥<4 and 2𝑥>2 𝑥<2 and 𝑥>1 1<𝑥<2

Solve the following Absolute Value inequalities with a > or ≥ sign 1. 𝑥−7 ≥5 → 𝑥−7≥5 or 𝑥−7≤−5 𝑥≥12 or 𝑥≤2 2. 2−5𝑥 >3 → 2−5𝑥>3 or 2−5𝑥<−3 −5𝑥>1 or −5𝑥<−5 𝑥<− 1 5 or 𝑥>1

Solve the Absolute Value Inequality 𝑥−3 2 ≥4 𝑥−3 2 ≥4 or 𝑥−3 2 ≤−4 𝑥−3≥8 or 𝑥−3≤−8 𝑥≥11 or 𝑥≤−5

Solve the Absolute Value inequality 𝑥+6 +3>17 Get the absolute value expression by itself: 𝑥+6 >14 𝑥+6>14 or 𝑥+6<−14 𝑥>8 or 𝑥<−20

Solve the Absolute Value Inequality 3|𝑥+8|≥6 Divide both sides by 3 first: |𝑥+8|≥2 𝑥+8≥2 or 𝑥+8≤−2 𝑥≥−6 or 𝑥≤−10

Absolute Value Inequalities with No Solution or Infinitely many solutions Example with No Solution: 𝑥−2 −6<−9 𝑥−2 <−3 Absolute value is never negative. Therefore, it can’t be less than -3. Thus, no solution. Example with Infinitely Many Solutions: 𝑥+2 >−3 Since absolute value by definition is always greater than or equal to zero, the above inequality is always true no matter what value is used for 𝑥.