Solving One-Variable Inequalities
24 questions
Question 21Question →
What is the correct solution set for the inequality shown below?
4(x−1)−2(x+3)≥5x+2
x≤−4
x≥−4
x≤0
x≤4
Show answer & explanation
Answer: x≤−4
Answer
The correct solution set is x≤−4.
The correct answer is derived by carefully distributing the coefficients, combining like terms, isolating the variable x, and applying the rule to reverse the inequality symbol when dividing by a negative number.
Step-by-Step Solution
1
Distribute the values 4 and −2 into their respective parentheses on the left side.
4x−4−2x−6≥5x+2
This step removes the parentheses and prepares the terms for grouping.
2
Combine the like terms on the left side of the inequality.
2x−10≥5x+2
Simplifying expressions on each side makes it easier to isolate the variable.
3
Subtract 5x from both sides, and add 10 to both sides.
−3x≥12
This moves all variable terms to the left side and all constant terms to the right side.
4
Divide both sides by −3 and reverse the direction of the inequality symbol.
x≤−4
When dividing or multiplying an inequality by a negative number, the relationship between the two sides inverses, so the sign must be flipped.
Key Concept
Solving one-variable linear inequalities requires isolating the variable using inverse operations, and reversing the inequality symbol when multiplying or dividing by a negative number.
Question 22Question →
Consider the following linear inequality:
3(x−2)−7x≥2(x+3)
Which of the following represents the correct solution set for x?
x≤−2
x≥−2
x≤0
x≥6
Show answer & explanation
Answer: x≤−2
Answer
The correct solution is x≤−2.
To find the solution, expand the terms to get 3x−6−7x≥2x+6. Combine the like terms on the left to form −4x−6≥2x+6. Subtracting 2x and adding 6 to both sides isolates the variable, resulting in −6x≥12. Finally, divide by −6. Because we are dividing by a negative number, the inequality sign must flip, giving the correct answer of x≤−2.
Step-by-Step Solution
1
Distribute the constants on both sides of the inequality.
3x−6−7x≥2x+6
Expanding the expressions clears the parentheses and allows us to combine like terms.
2
Combine the like terms on the left side of the inequality.
−4x−6≥2x+6
3x−7x simplifies to −4x.
3
Move all terms containing x to one side and constants to the other side.
−6x≥12
Subtracting 2x from both sides gives −6x−6≥6. Then, adding 6 to both sides isolates the variable term.
4
Divide both sides by −6 and reverse the inequality symbol.
x≤−2
When multiplying or dividing an inequality by a negative number, the inequality direction must be flipped to remain mathematically true.
Key Concept
Solving Linear Inequalities with Variables on Both Sides
Question 23Question →
Which of the following describes all values of x that satisfy the inequality below?
2(x−4)−5x<x+4
x>−3
x<−3
x>3
x>1
Show answer & explanation
Answer: x>−3
Answer
The correct solution is x>−3.
The correct answer accurately simplifies the equation down to −4x<12. Because isolating the variable in the final step requires dividing by −4, the less than symbol (<) must be reversed to a greater than symbol (>), resulting in x>−3.
Step-by-Step Solution
1
Distribute the 2 into the parentheses on the left side.
2x−8−5x<x+4
To clear the parentheses and access the terms inside.
2
Combine the like terms (2x and −5x) on the left side.
−3x−8<x+4
To simplify the left side of the inequality.
3
Subtract x from both sides to group the variable terms.
−4x−8<4
To collect all terms containing x on one side.
4
Add 8 to both sides of the inequality.
−4x<12
To isolate the term with the variable.
5
Divide both sides by -4 and reverse the inequality sign.
x>−3
When multiplying or dividing an inequality by a negative number, the inequality symbol must be flipped to maintain a mathematically true statement.
Key Concept
Solving linear inequalities with variables on both sides requires combining like terms and reversing the inequality symbol when dividing by a negative coefficient.
Question 24Question →
What is the minimum integer value of x that satisfies the inequality shown below?
3(2x−5)−8x≤4(x−1)−23
Show answer & explanation
Answer: 2
Answer
2
Solving the inequality yields x≥2. Since x must be greater than or equal to 2, the minimum integer value that makes this statement true is exactly 2.
Step-by-Step Solution
1
Distribute the constants into the parentheses on both sides of the inequality.
6x−15−8x≤4x−4−23
This removes the parentheses, allowing us to combine like terms.
2
Combine the like terms on each side of the inequality.
−2x−15≤4x−27
Simplifying each side makes it easier to isolate the variable.
3
Subtract 4x from both sides to group the variable terms on the left side.
−6x−15≤−27
We need all terms with x on one side of the inequality to solve for it.
4
Add 15 to both sides to isolate the term with the variable.
−6x≤−12
This leaves only the −6x term on the left side.
5
Divide both sides by −6, remembering to flip the inequality symbol.
x≥2
Dividing or multiplying an inequality by a negative number reverses the direction of the inequality sign.
6
Determine the minimum integer value that satisfies the simplified inequality.
2
The inequality states that x must be greater than or equal to 2. The smallest integer that meets this condition is 2.
Key Concept
Solving linear inequalities requiring the distributive property, combining like terms, and reversing the inequality symbol when dividing by a negative coefficient.
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