Questions: 5x - 14 < 8x + 4 5x + 34 = - 2 + 14x 5 - 7x ≤ 3x + 45 - 26 + 6x = 2x + 34

5x - 14 < 8x + 4 5x + 34 = - 2 + 14x 5 - 7x ≤ 3x + 45 - 26 + 6x = 2x + 34
Transcript text: 5x - 14 < 8x + 4 5x + 34 = - 2 + 14x 5 - 7x ≤ 3x + 45 - 26 + 6x = 2x + 34
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Solution

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Solution Steps

To solve these inequalities and equations, we will isolate the variable \( x \) on one side of each equation or inequality. This involves using basic algebraic operations such as addition, subtraction, multiplication, and division.

  1. For the inequality \( 5x - 14 < 8x + 4 \), we will move all terms involving \( x \) to one side and constant terms to the other side, then solve for \( x \).

  2. For the equation \( 5x + 34 = -2 + 14x \), we will similarly move all terms involving \( x \) to one side and constant terms to the other side, then solve for \( x \).

  3. For the inequality \( 5 - 7x \leq 3x + 45 \), we will again move all terms involving \( x \) to one side and constant terms to the other side, then solve for \( x \).

Step 1: Solve the Inequality \( 5x - 14 < 8x + 4 \)

To solve the inequality, we rearrange it as follows: \[ 5x - 14 < 8x + 4 \implies -14 - 4 < 8x - 5x \implies -18 < 3x \implies x > -6 \] Thus, the solution to the inequality is: \[ x > -6 \quad \text{or} \quad (-6 < x < \infty) \]

Step 2: Solve the Equation \( 5x + 34 = -2 + 14x \)

Rearranging the equation gives: \[ 5x + 34 = -2 + 14x \implies 34 + 2 = 14x - 5x \implies 36 = 9x \implies x = 4 \] Thus, the solution to the equation is: \[ x = 4 \]

Step 3: Solve the Inequality \( 5 - 7x \leq 3x + 45 \)

Rearranging the inequality gives: \[ 5 - 7x \leq 3x + 45 \implies 5 - 45 \leq 3x + 7x \implies -40 \leq 10x \implies x \geq -4 \] Thus, the solution to the inequality is: \[ x \geq -4 \quad \text{or} \quad (-4 \leq x < \infty) \]

Final Answer

The solutions to the problems are:

  1. \( x > -6 \)
  2. \( x = 4 \)
  3. \( x \geq -4 \)

In boxed form: \[ \boxed{x > -6}, \quad \boxed{x = 4}, \quad \boxed{x \geq -4} \]

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