Questions: Efficiently Solving Inequalities Let's solve more complicated inequalities. Warm-up Lots of Negatives Here is an inequality: -x >= -4 (1) Without testing any values, predict what the solutions to this inequality will look like on the number line. - Where will the starting (or boundary) point be? - Will the circle at that point be filled or unfilled? Will the shading go to the right or to the left of that point? (2) Test each value to see whether it is a solution to the inequality -x >= -4. a. 3 b. -3 c. 4 d. -4 e. 4.001 f. -4.001 (3) Graph all possible solutions to the inequality on the number line:

Efficiently Solving Inequalities Let's solve more complicated inequalities. Warm-up Lots of Negatives Here is an inequality: -x >= -4 (1) Without testing any values, predict what the solutions to this inequality will look like on the number line. - Where will the starting (or boundary) point be? - Will the circle at that point be filled or unfilled? Will the shading go to the right or to the left of that point? (2) Test each value to see whether it is a solution to the inequality -x >= -4. a. 3 b. -3 c. 4 d. -4 e. 4.001 f. -4.001 (3) Graph all possible solutions to the inequality on the number line:
Transcript text: Efficiently Solving Inequalities Let's solve more complicated inequalities. Warm-up Lots of Negatives Here is an inequality: $-x \geq-4$ (1) Without testing any values, predict what the solutions to this inequality will look like on the number line. - Where will the starting (or boundary) point be? -Will the circle at that point be filled or unfilled? $\qquad$ Will the shading go to the right or to the left of that point? $\qquad$ (2) Test each value to see whether it is a solution to the inequality $-x \geq-4$. a. 3 b. -3 C. 4 d. -4 e. 4.001 f. -4.001 (3) Graph all possible solutions to the inequality on the number line: W10R1-UNTT 6. SECTION G I LESSON 15
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Solution

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Solution Steps

Step 1: Predicting the Solution Set

  1. Boundary Point: The boundary point will be at x = 4. This is because if we multiply both sides of the inequality by -1 (and flip the inequality sign), we get \(x \leq 4\).

  2. Circle Type: The circle at x = 4 will be filled because the inequality includes "equal to" (\(\leq\)).

  3. Shading Direction: The shading will go to the left of the boundary point because the inequality states that x is less than or equal to 4.

Step 2: Testing Values
  • a. \(x = 3\): \(-3 \geq -4\) (True)
  • b. \(x = -3\): \(-(-3) \geq -4 \Rightarrow 3 \geq -4\) (True)
  • c. \(x = 4\): \(-4 \geq -4\) (True)
  • d. \(x = -4\): \(-(-4) \geq -4 \Rightarrow 4 \geq -4\) (True)
  • e. \(x = 4.001\): \(-4.001 \geq -4\) (False)
  • f. \(x = -4.001\): \(-(-4.001) \geq -4 \Rightarrow 4.001 \geq -4\) (True)
Step 3: Graphing the Solution

The graph should be a number line with a closed (filled) circle at 4, and shading extending to the left towards negative infinity.

Final Answer

The solution to the inequality \(-x \geq -4\) is \(x \leq 4\). The graph would be a filled-in circle at 4, with the line shaded to the left.

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