Examkin

Question

Difficulty: MediumLinear Inequalities in One Variable

Which inequality is equivalent to −3(3x−5)−4<−2x+18-3(3x - 5) - 4 < -2x + 18?

  1. A
    x<−1x < -1
  2. B
    x>−377x > -\frac{37}{7}
  3. x>−1x > -1Answer
  4. D
    x>−297x > -\frac{29}{7}

Answer

x>−1x > -1
The correct inequality is found by first expanding −3(3x−5)-3(3x - 5) to get −9x+15-9x + 15. The inequality becomes −9x+15−4<−2x+18-9x + 15 - 4 < -2x + 18, which simplifies to −9x+11<−2x+18-9x + 11 < -2x + 18. Adding 2x2x to both sides gives −7x+11<18-7x + 11 < 18. Subtracting 1111 from both sides gives −7x<7-7x < 7. Dividing both sides by the negative coefficient −7-7 requires reversing the inequality sign, yielding x>−1x > -1.

Step-by-Step Solution

1
Distribute −3-3 to both terms inside the parentheses: −3(3x−5)-3(3x - 5).
−9x+15−4<−2x+18-9x + 15 - 4 < -2x + 18
This expands the expression using the distributive property, making sure to multiply both 3x3x and −5-5 by −3-3, which changes the sign of the constant term to positive 1515.
2
Combine the constant terms on the left side of the inequality.
−9x+11<−2x+18-9x + 11 < -2x + 18
Simplifying the constant terms (15−4=1115 - 4 = 11) makes it easier to isolate the variable.
3
Add 2x2x to both sides of the inequality to group the variable terms on the left.
−7x+11<18-7x + 11 < 18
This combines the xx terms on one side of the inequality.
4
Subtract 1111 from both sides of the inequality to group the constant terms on the right.
−7x<7-7x < 7
This isolates the term with the variable on the left side.
5
Divide both sides of the inequality by −7-7 and reverse the inequality sign.
x>−1x > -1
Since we are dividing by a negative number, the inequality sign must be reversed from less than (<<) to greater than (>>).

Key Concept

Solving linear inequalities in one variable, including the distributive property and division by negative numbers.
Rate this question

Topics

Linear Inequalities in One Variable

More questions from this topic