Solve the compound inequality: -2x + 8 > 0 or -12 ≥ -2x + 8
Understand the Problem
The question provides two inequalities involving the variable 'x'. The task is to solve each inequality and express the solution set. These are linear inequalities, which can be solved by isolating 'x' on one side. The presence of 'or' indicates that the solution set includes all values of 'x' that satisfy either inequality.
Answer
$x \geq -6$ or $x < -2$
Answer for screen readers
$x \geq -6$ or $x < -2$
Steps to Solve
- Solve the first inequality Isolate $x$ in the inequality $2x + 7 \geq -5$:
$2x + 7 \geq -5$ $2x \geq -5 - 7$ $2x \geq -12$ $x \geq -6$
- Solve the second inequality Isolate $x$ in the inequality $-4x - 5 > 3$:
$-4x - 5 > 3$ $-4x > 3 + 5$ $-4x > 8$ $x < -2$ (Note: Dividing/multiplying by a negative number flips the inequality sign)
- Express the solution set The solution set includes all $x$ such that $x \geq -6$ or $x < -2$.
$x \geq -6$ or $x < -2$
More Information
The solution includes all real numbers because any real number is either greater than or equal to $-6$ or it is less than $-2$. The interval between $-6$ and $-2$ is included in both inequalities.
Tips
A common mistake is forgetting to flip the inequality sign when multiplying or dividing by a negative number. Another mistake is misinterpreting the 'or' condition, which means either inequality can be true, and the solution is the union of the two individual solution sets.
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