# Solving Inequalities

I. One Step Inequalities

1. x + 4 < 5 2. x - 5 ≥ -10 3.

x < 1 x ≥ -5 x > 4 x ≤ 24

5. -

-3 -3 flip the inequality

x ≤ -5 x > -10

A note of caution__4x__>__20__4. (2) $\frac{x}{2}$ ≤ (2)__-4 -4____+5 +5__4 4x < 1 x ≥ -5 x > 4 x ≤ 24

5. -

__3x__≥__15__6. (-2) $\frac{x}{-2}$ < 5 (-2) flip the inequality-3 -3 flip the inequality

x ≤ -5 x > -10

7.

22

8. (4)$\frac{x}{4}$ ≥ -8 (4) Do you flip???

II. Two step Inequalities__2x__>__-12__Do you flip????**NO**The number you would divide by is positive22

8. (4)$\frac{x}{4}$ ≥ -8 (4) Do you flip???

**NO**The number you multiply by is positive
1. 3x + 4 > 10 2. -4x + 5 ≥ 21

3 3 -4 -4 dividing by a negative

x > 2 x ≤ -4 FLIP the inequality symbol

Challenge 2 > 6 - x flip entire inequality so x is on the left

6 - x < 2 now solve

-1 -1 dividing by a negative number!!!

x > 4 Flip the inequality symbol

III. Multi-step Inequalities__-4 -4____-5 -5____3x__>__6____-4x__≥__16__3 3 -4 -4 dividing by a negative

x > 2 x ≤ -4 FLIP the inequality symbol

Challenge 2 > 6 - x flip entire inequality so x is on the left

6 - x < 2 now solve

__-6 -6____- x__<__-4__-1 -1 dividing by a negative number!!!

x > 4 Flip the inequality symbol

1. 6(x + 1) > 6

6x + 6 > 6 distributive property

6x > 0 subtract 6 from both sides

X > 0 divide both sides by 6

2. 10 - 11x > -5x -4

10 - 6x > -4 subtract (-5x) from both sides

-6x > -14 subtract 10 from both sides

X < 7/3 divide by (-6) and flip the inequality symbol

IV. Special Cases6x + 6 > 6 distributive property

6x > 0 subtract 6 from both sides

X > 0 divide both sides by 6

2. 10 - 11x > -5x -4

10 - 6x > -4 subtract (-5x) from both sides

-6x > -14 subtract 10 from both sides

X < 7/3 divide by (-6) and flip the inequality symbol

1. 7x - 11x + 3 ≥ 3 - 4x

-4x + 3 ≥ 3 - 4x combine like terms

3 ≥ 3 this is a true statement because 3 = 3

The solution is

2. 4(3-2x) > 2(6 - 4x)

12 - 8x > 12 -8x distributive property

12 > 12 this is a false statement - 12 is not greater than 12

There is

-4x + 3 ≥ 3 - 4x combine like terms

__+4x +4x__add 4x to both sides3 ≥ 3 this is a true statement because 3 = 3

The solution is

**All Real Numbers**2. 4(3-2x) > 2(6 - 4x)

12 - 8x > 12 -8x distributive property

__+8x +8x__add (8x) to both sides12 > 12 this is a false statement - 12 is not greater than 12

There is

**No Solution****. You can solve inequalities just like solving an equation. Be cautious when solving inequalities that you make sure to flip the inequality symbol when you multiply or divide by a negative number. If the variable is eliminated from the equation then you will have a special case. If you get a true statement then the solution is "all real numbers". If you get a false statement then the solution is "no solution".**

*Let's wrap it up*
Related Links:Math algebra Dependent and Independent Variables Measures of Variability : Mean Absolute Deviation |

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