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Questions and Answers
What is the first step in solving the inequality $3x + 1 < \frac{4x}{5} + 3$ algebraically?
What is the first step in solving the inequality $3x + 1 < \frac{4x}{5} + 3$ algebraically?
- Convert $\frac{4x}{5}$ into a decimal
- Multiply both sides by 5
- Subtract $3x$ from both sides (correct)
- Add 1 to both sides
What is the equivalent form of the inequality after simplifying $3x + 1 < \frac{4x}{5} + 3$?
What is the equivalent form of the inequality after simplifying $3x + 1 < \frac{4x}{5} + 3$?
- $15x < 14$
- $x < 4$
- $5x + 1 < 4x + 15$
- $-2x < 14$ (correct)
What is the solution in interval notation for the inequality $3x + 1 < \frac{4x}{5} + 3$?
What is the solution in interval notation for the inequality $3x + 1 < \frac{4x}{5} + 3$?
- (7, ∞)
- (-∞, 7)
- (-∞, 4) (correct)
- (7, 4)
Which graphical representation accurately depicts the solution of $3x + 1 < \frac{4x}{5} + 3$?
Which graphical representation accurately depicts the solution of $3x + 1 < \frac{4x}{5} + 3$?
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What occurs if both sides of the inequality $3x + 1 < \frac{4x}{5} + 3$ are multiplied by -1?
What occurs if both sides of the inequality $3x + 1 < \frac{4x}{5} + 3$ are multiplied by -1?
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Study Notes
Inequality Problem Overview
- Inequality: (3x + 1 < \frac{4x}{5} + 3)
- Objective: Solve both algebraically and graphically.
Algebraic Solution Steps
- Start with the original inequality: (3x + 1 < \frac{4x}{5} + 3).
- Eliminate the fraction by multiplying the entire inequality by 5: (15x + 5 < 4x + 15).
- Rearrange to isolate (x): (15x - 4x < 15 - 5).
- Simplify to: (11x < 10).
- Divide by 11: (x < \frac{10}{11}).
Interval Notation
- The solution in interval notation: ((- \infty, \frac{10}{11})).
Graphical Representation
- Graph the solution on a number line.
- Shade to the left of (\frac{10}{11}) indicating all values less than (\frac{10}{11}).
- Use an open circle at (\frac{10}{11}) to denote that it's not included in the solution set.
Possible Graph Options
- Analyze and select the correct graph representation from given options (A through F).
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