Understand Solving Equations and Inequalities
Help Questions
6th Grade Math › Understand Solving Equations and Inequalities
From the set ${2,3,4,5}$, which values make the inequality $x<4$ true? (Test each value.)
${2,3,4}$
${2,3,4,5}$
${2,3}$
${4,5}$
Explanation
This question tests understanding of solving inequalities as finding which values from a specified set make the statement true, using substitution to check each value. The solving process involves the given inequality x<4 and the test set {2,3,4,5}, where you determine which values satisfy it by substitution: pick a value like x=3, replace (3<4), evaluate and check if true—yes, so x=3 is a solution; inequalities can have multiple solutions from the set. For example, with inequality x<4 and set {2,3,4,5}, check x=2:2<4 true, x=3:3<4 true, x=4:4<4 false, x=5:5<4 false, so solutions are {2,3}. Correct solution identification comes from testing each and confirming strict inequality. Common errors include including x=4 when < excludes equality, or not testing all values. Testing systematically: (1) list the set {2,3,4,5}, (2) test each like x=4:4<4 false, (3) mark true values like x=2 and x=3, (4) collect solutions {2,3}. Remember < means less than, not less than or equal.
Consider the equation $$2x - 3 = x + 1$$. If you solve this algebraically, you get $$x = 4$$. Now test whether each value from the set {$$2, 3, 4, 5$$} actually satisfies the original equation by substitution. What do you discover?
Values $$x = 3$$ and $$x = 4$$ both satisfy the equation, suggesting the algebraic solution missed one possibility
Only $$x = 4$$ satisfies the equation, confirming the algebraic solution, while other values create false statements
Values $$x = 4$$ and $$x = 5$$ both satisfy the equation, confirming that linear equations can have multiple solutions
All values except $$x = 2$$ satisfy the equation, indicating an error in the algebraic solution process
Explanation
Testing by substitution: For $$x = 2$$: $$2(2) - 3 = 1$$ and $$2 + 1 = 3$$, so $$1 = 3$$ (false). For $$x = 3$$: $$2(3) - 3 = 3$$ and $$3 + 1 = 4$$, so $$3 = 4$$ (false). For $$x = 4$$: $$2(4) - 3 = 5$$ and $$4 + 1 = 5$$, so $$5 = 5$$ (true). For $$x = 5$$: $$2(5) - 3 = 7$$ and $$5 + 1 = 6$$, so $$7 = 6$$ (false). Only $$x = 4$$ works, confirming the algebraic solution.
For which value from the set {$$4, 5, 6, 7$$} does the equation $$2(x - 3) = x + 1$$ become false when you substitute it?
Values $$x = 5, 6, 7$$ all make the equation false when properly substituted
Only $$x = 4$$ makes the equation false since substitution gives $$2 = 5$$
Values $$x = 4, 5, 6$$ all make the equation false when properly substituted
Only $$x = 7$$ makes the equation false since substitution gives $$8 = 8$$
Explanation
Substituting each value: For $$x = 4$$: $$2(4-3) = 2(1) = 2$$ and $$4 + 1 = 5$$, so $$2 = 5$$ (false). For $$x = 5$$: $$2(5-3) = 4$$ and $$5 + 1 = 6$$, so $$4 = 6$$ (false). For $$x = 6$$: $$2(6-3) = 6$$ and $$6 + 1 = 7$$, so $$6 = 7$$ (false). For $$x = 7$$: $$2(7-3) = 8$$ and $$7 + 1 = 8$$, so $$8 = 8$$ (true). Only $$x = 7$$ makes the equation true, so $$x = 4, 5, 6$$ all make it false.
From the set {$$-1, 0, 1, 2$$}, which value(s) make the inequality $$x^2 + 1 ≤ 2$$ true when substituted?
Only $$x = 0$$ satisfies the inequality since it's the only value that makes $$x^2 = 0$$
Only $$x = -1$$ and $$x = 1$$ satisfy the inequality when their squares are computed and substituted
Values $$x = -1, 0,$$ and $$1$$ all satisfy the inequality when substituted and computed correctly
All four values from the set satisfy the inequality since squaring always produces small positive results
Explanation
Substituting each value into $$x^2 + 1 ≤ 2$$: For $$x = -1$$: $$(-1)^2 + 1 = 1 + 1 = 2$$, and $$2 ≤ 2$$ is true. For $$x = 0$$: $$0^2 + 1 = 1$$, and $$1 ≤ 2$$ is true. For $$x = 1$$: $$1^2 + 1 = 2$$, and $$2 ≤ 2$$ is true. For $$x = 2$$: $$2^2 + 1 = 5$$, and $$5 ≤ 2$$ is false. So $$x = -1, 0, 1$$ all work.
A student is testing whether values from {$$1, 2, 3, 4$$} satisfy the compound inequality $$2 ≤ x + 1 < 5$$. After substitution, which values from the set make both parts of the compound inequality true?
Values $$2, 3,$$ and $$4$$ all satisfy both inequality conditions when substituted and evaluated
Values $$1, 2,$$ and $$3$$ all satisfy both inequality conditions when substituted and evaluated
Only values $$2$$ and $$3$$ satisfy both inequality conditions when substituted and evaluated
Only values $$1$$ and $$4$$ satisfy both inequality conditions when substituted and evaluated
Explanation
For $$2 ≤ x + 1 < 5$$, substituting each value: For $$x = 1$$: $$2 ≤ 1 + 1 < 5$$ becomes $$2 ≤ 2 < 5$$ (true). For $$x = 2$$: $$2 ≤ 2 + 1 < 5$$ becomes $$2 ≤ 3 < 5$$ (true). For $$x = 3$$: $$2 ≤ 3 + 1 < 5$$ becomes $$2 ≤ 4 < 5$$ (true). For $$x = 4$$: $$2 ≤ 4 + 1 < 5$$ becomes $$2 ≤ 5 < 5$$ (false, since $$5 < 5$$ is false). Therefore, $$x = 1, 2, 3$$ all work.
A teacher asks students to find which values from {$$0, 1, 2, 3$$} satisfy $$\frac{x + 4}{2} = x + 1$$. One student claims that $$x = 2$$ works because $$\frac{2 + 4}{2} = 3$$ and $$2 + 1 = 3$$, so both sides equal $$3$$. Is this student correct, and what about the other values?
The student is correct about $$x = 2$$, and additionally $$x = 0$$ also satisfies the equation when checked
The student made an error since $$\frac{2 + 4}{2} = 4$$, not $$3$$, so $$x = 2$$ doesn't actually work
The student is correct about $$x = 2$$, and both $$x = 1$$ and $$x = 3$$ also work when substituted properly
The student is correct about $$x = 2$$, but it's the only value from the set that satisfies the equation
Explanation
The student's work for $$x = 2$$ is correct: $$\frac{2 + 4}{2} = \frac{6}{2} = 3$$ and $$2 + 1 = 3$$, so $$3 = 3$$ is true. Checking other values: For $$x = 0$$: $$\frac{0 + 4}{2} = 2$$ and $$0 + 1 = 1$$, so $$2 = 1$$ (false). For $$x = 1$$: $$\frac{1 + 4}{2} = 2.5$$ and $$1 + 1 = 2$$, so $$2.5 = 2$$ (false). For $$x = 3$$: $$\frac{3 + 4}{2} = 3.5$$ and $$3 + 1 = 4$$, so $$3.5 = 4$$ (false). Only $$x = 2$$ works.
Sarah claims that $$x = 3$$ is a solution to the inequality $$5x - 7 ≤ 2x + 2$$ because when she substitutes, she gets $$8 ≤ 8$$, which is true. However, when she solves the inequality algebraically, she gets $$x ≤ 3$$. Which statement best explains this situation?
Sarah made an error in her algebraic solution since substitution shows $$x = 3$$ works perfectly
Sarah made an error in substitution since $$5(3) - 7 = 7$$, not $$8$$ as she calculated
Sarah's substitution confirms that $$x = 3$$ is indeed a solution, and it's the boundary value
Sarah's algebraic work is wrong because inequalities cannot have boundary values like $$x = 3$$
Explanation
Sarah's work is completely correct. When $$x = 3$$: $$5(3) - 7 = 15 - 7 = 8$$ and $$2(3) + 2 = 8$$, so $$8 ≤ 8$$ is true. Solving algebraically: $$5x - 7 ≤ 2x + 2$$, so $$3x ≤ 9$$, thus $$x ≤ 3$$. The value $$x = 3$$ is the boundary value where equality holds. Choice A is wrong because both methods agree. Choice C is wrong because $$5(3) - 7 = 8$$. Choice D is wrong because inequalities can include boundary values.
A game gives bonus points based on $x$. From the set ${3,4,5,6}$, which values make the equation $x+2=8$ true?
${5}$
${6}$
${4,5}$
No values in the set make the equation true.
Explanation
This question tests understanding of solving equations as finding which values from a specified set make the statement true, using substitution to check each value. The solving process involves the given equation x + 2 = 8 and the test set {3,4,5,6}, where you determine which values satisfy it by substituting each one. For example, try x=6:6+2=8=8 true; equations typically have one solution. For instance, x=3:3+2=5≠8 false, x=4:6≠8 false, x=5:7≠8 false, x=6:8=8 true, so {6}. The correct solution is identified by substitution as {6}. A common error is wrong value like x=5 when 7≠8, or not testing all. To test systematically: (1) list {3,4,5,6}, (2) substitute each into x+2=8, (3) mark true x=6, (4) collect {6}.
Consider the inequality $$4x - 5 > 7$$. If you test the values {$$2, 3, 4, 5$$} by substitution, which statement correctly describes what you would find?
Value $$3$$ makes the inequality an equality, while $$4$$ and $$5$$ satisfy the strict inequality condition
Values $$3, 4,$$ and $$5$$ satisfy the inequality, while only $$2$$ creates a false statement when substituted
Values $$4$$ and $$5$$ satisfy the inequality, while $$2$$ and $$3$$ create false statements when substituted
All four values satisfy the inequality since each substitution results in a value greater than $$7$$
Explanation
Testing each value in $$4x - 5 > 7$$: For $$x = 2$$: $$4(2) - 5 = 3$$, and $$3 > 7$$ is false. For $$x = 3$$: $$4(3) - 5 = 7$$, and $$7 > 7$$ is false. For $$x = 4$$: $$4(4) - 5 = 11$$, and $$11 > 7$$ is true. For $$x = 5$$: $$4(5) - 5 = 15$$, and $$15 > 7$$ is true. So only $$x = 4$$ and $$x = 5$$ satisfy the inequality. Choice C is wrong because $$x = 3$$ gives equality for $$4x - 5 = 7$$, not $$4x - 5 > 7$$.
Marcus is checking which values from the set {$$-2, -1, 0, 1, 2$$} make the inequality $$3x + 4 > x - 2$$ true. He substitutes each value and gets these results: For $$x = -2$$: $$-2 > -6$$ (true), For $$x = -1$$: $$1 > -3$$ (true), For $$x = 0$$: $$4 > -2$$ (true), For $$x = 1$$: $$7 > -1$$ (true), For $$x = 2$$: $$10 > 0$$ (true). What error did Marcus make?
He simplified $$3x + 4$$ incorrectly when $$x$$ was negative in the first two cases
He evaluated $$x - 2$$ incorrectly, writing $$-6$$ instead of $$-4$$ when $$x = -2$$
He substituted the values into the wrong side of the inequality in each case
He substituted the values into the original inequality instead of solving it first algebraically
Explanation
When $$x = -2$$, Marcus should get $$x - 2 = -2 - 2 = -4$$, not $$-6$$. The correct comparison should be $$-2 > -4$$ (true). Marcus made an arithmetic error. Choice A is wrong because he substituted correctly into both sides. Choice B is wrong because $$3(-2) + 4 = -2$$ and $$3(-1) + 4 = 1$$ are correct. Choice C is wrong because substitution is the correct method for this problem.