Answered

Solve. Express the answer as an inequality statement, an equality, or the empty set.


  1. $x>5$ and $x \leq-2$
  2. $x<-1$ and $x \geq 4$
  3. $2 x>-6$ and $x \leq 1$
  4. $-5 x+13>-7$ and $4 x-6 \leq 10$
  5. $x-3 \leq 4 x$ and $2 x+6<18$
  6. $9 x+2<-7$ and $-2 x-8 \leq 12$

Verified Answer :

  • To solve systems of linear inequalities, we find the values of \(x\) that satisfy both inequalities at the same time.
  • For systems with conflicting conditions (like \(x > 5\) and \(x \leq -2\)), the solution is the empty set.
  • For systems where the conditions overlap (like \(x > -3\) and \(x \leq 1\)), the solution is an interval, such as \(-3 < x \leq 1\).
  • The final solutions are: 10) \(\emptyset\), 11) \(\emptyset\), 13) \(-3 < x \leq 1\), 14) \(x < 4\), 15) \(-1 \leq x < 6\), and 16) \(-10 \leq x < -1\).

Explanation

  1. Problem Analysis and Strategy

To solve a system of linear inequalities, we need to find the intersection of the solution sets of the individual inequalities. For each system, we solve the two inequalities separately to determine the possible values of \(x\) that satisfy both simultaneously. If no such values exist, the result is the empty set.

  1. Solving Systems 10 and 11
  2. For the system \(x > 5\) and \(x \leq -2\), we have:
  • The first inequality \(x > 5\) covers all numbers greater than 5.
  • The second inequality \(x \leq -2\) covers all numbers less than or equal to -2.

Since there is no number that is simultaneously greater than 5 and less than or equal to -2, the solution set is the empty set.

  1. For the system \(x < -1\) and \(x \geq 4\), we have:
  • The first inequality \(x < -1\) covers all numbers less than -1.
  • The second inequality \(x \geq 4\) covers all numbers greater than or equal to 4.

Since there is no number that is simultaneously less than -1 and greater than or equal to 4, the solution set is the empty set.

  1. Solving Systems 13 and 14
  2. For the system \(2x > -6\) and \(x \leq 1\):
  • Solve the first: \(2x > -6 \Rightarrow x > -3\).
  • The second is already solved: \(x \leq 1\).
  • Combining these, we look for \(x\) such that \(-3 < x \leq 1\).
  1. For the system \(-5x + 13 > -7\) and \(4x - 6 \leq 10\):
  • Solve the first: \(-5x > -20 \Rightarrow x < 4\) (remembering to flip the inequality sign when dividing by a negative number).
  • Solve the second: \(4x \leq 16 \Rightarrow x \leq 4\).
  • Combining these, both conditions are satisfied when \(x < 4\).
  1. Solving Systems 15 and 16
  2. For the system \(x - 3 \leq 4x\) and \(2x + 6 < 18\):
  • Solve the first: \(-3 \leq 3x \Rightarrow -1 \leq x\) (or \(x \geq -1\)).
  • Solve the second: \(2x < 12 \Rightarrow x < 6\).
  • Combining these, both conditions are satisfied when \(-1 \leq x < 6\).
  1. For the system \(9x + 2 < -7\) and \(-2x - 8 \leq 12\):
  • Solve the first: \(9x < -9 \Rightarrow x < -1\).
  • Solve the second: \(-2x \leq 20 \Rightarrow x \geq -10\) (again, flipping the inequality sign when dividing by -2).
  • Combining these, both conditions are satisfied when \(-10 \leq x < -1\).
  1. Final Results

The solutions for the systems are:

  1. \(\emptyset\) (Empty set)
  2. \(\emptyset\) (Empty set)
  3. \(-3 < x \leq 1\)
  4. \(x < 4\)
  5. \(-1 \leq x < 6\)
  6. \(-10 \leq x < -1\)

These results represent the intervals on the number line where both given conditions are satisfied.

Examples

Imagine you are a quality control manager in a factory. A part you are manufacturing must have a specific weight range. For instance, the first test requires the weight to be greater than 10 grams, and the second test requires the weight to be at most 15 grams. Finding the acceptable weight range for the part is equivalent to solving a system of two linear inequalities: \(w > 10\) and \(w \leq 15\), which gives the solution \(10 < w \leq 15\). This ensures that all components meeting these strict criteria perform perfectly in their final assembly.

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