Mixed Questions Set: Solving Equations & Inequalities

Ans: (b)
Explanation: To solve \(2x + 5 = 13\), first subtract 5 from both sides: \(2x = 8\). Then divide both sides by 2: \(x = 4\). Option (a) comes from incorrectly subtracting 5 and then dividing by 5. Option (c) results from dividing by 2 before subtracting 5. Option (d) comes from adding instead of subtracting.

Ans: (c)
Explanation: The phrase "less than or equal to" translates to the symbol \(\leq\). So the inequality is \(x \leq 10\). Option (a) represents "greater than," option (b) represents "greater than or equal to," and option (d) represents "strictly less than" (without the equal part).

Ans: (b)
Explanation: To solve, collect variable terms on one side: \(3x - 2x = 4 + 7\), which gives \(x = 11\). This is a two-step rearrangement. Option (a) results from incorrectly adding instead of subtracting. Option (c) comes from a sign error. Option (d) results from dividing by the wrong coefficient.

Ans: (a)
Explanation: Divide both sides by 5: \(\frac{5x}{5} > \frac{20}{5}\), which gives \(x > 4\). The inequality sign stays the same because we divide by a positive number. Option (b) comes from dividing 20 by 4 instead of 5. Option (c) incorrectly reverses the inequality sign. Option (d) uses the wrong inequality symbol.

Ans: (a)
Explanation: First, subtract 2 from both sides: \(\frac{x}{3} = 3\). Then multiply both sides by 3: \(x = 9\). Option (b) comes from forgetting to subtract 2 before multiplying. Option (c) results from only subtracting and not multiplying. Option (d) comes from dividing by 3 instead of multiplying.

Ans: add the same number to the other side (or equivalent phrasing such as "add it to both sides" or "apply the same operation to both sides")
Explanation: This tests understanding of the Property of Equality or Balancing Property-equations remain equal only when the same operation is performed on both sides.

Ans: greater than 5 (or "to the right of 5 on a number line," or "larger than 5")
Explanation: An inequality like \(x > 5\) defines a solution set-all values that make the inequality true. Here, that means every number greater than 5, typically shown as a ray on a number line.

Ans: divide, 2 (or "divide both sides by 2")
Explanation: Inverse operations are used to isolate the variable. Since \(2x\) means 2 times \(x\), division by 2 is the inverse operation needed to solve for \(x\).

Ans: reverse (or "flip" or "switch") the inequality sign
Explanation: This is the Inequality Reversal Property. For example, \(-2x > 6\) becomes \(x < -3\)="" after="" dividing="" by="" \(-2\).="" this="" property="" is="" unique="" to="" inequalities="" and="" does="" not="" apply="" to="" equations.="">

Ans: true (or "equal" or "balanced")
Explanation: A solution is defined as any value that, when substituted for the variable, makes the two sides of the equation equal. This is the fundamental definition students must understand for equation-solving.

Ans:
Let \(m\) = number of text messages.
Write the equation: The total bill is the base fee plus the cost per message times the number of messages.
\[15 + 0.10m = 23\]
Solve for \(m\): Subtract 15 from both sides.
\[0.10m = 23 - 15\]
\[0.10m = 8\]
Divide both sides by 0.10.
\[m = \frac{8}{0.10} = 80\]
Check: \(15 + 0.10(80) = 15 + 8 = 23\) ✓
Final Answer: 80 text messages were sent.

Ans:
Let \(w\) = width of the rectangle (in inches).
Then length = \(w + 4\) (in inches).
Write the perimeter equation: Perimeter = \(2l + 2w\)
\[2(w + 4) + 2w = 32\]
Simplify:
\[2w + 8 + 2w = 32\]
\[4w + 8 = 32\]
Subtract 8 from both sides:
\[4w = 24\]
Divide by 4:
\[w = 6 \text{ inches}\]
Find the length:
\[l = w + 4 = 6 + 4 = 10 \text{ inches}\]
Check: Perimeter = \(2(10) + 2(6) = 20 + 12 = 32\) ✓
Final Answer: Width = 6 inches; Length = 10 inches.

Ans:
Let \(T\) = temperature in °F.
Write the inequality: Since the temperature must stay above 32°F:
\[T > 32\]
Check if 28°F is safe: Substitute \(T = 28\):
\[28 > 32\]
This is false.
Final Answer: The inequality is \(T > 32\). A temperature of 28°F is not safe because it does not satisfy the inequality.

Ans:
A compound inequality contains two inequality signs. Solve by subtracting 1 from all three parts.
\[2 - 1 < x="" +="" 1="" -="" 1="">< 8="" -="" 1\]="">
\[1 < x="">< 7\]="">
Interpretation: \(x\) is any number strictly between 1 and 7 (not including 1 or 7).
Number line representation: Draw an open circle at 1 and an open circle at 7, with a line segment between them.
Final Answer: \(1 < x="">< 7\).="" on="" a="" number="" line:="" open="" circle="" at="" 1,="" line="" to="" open="" circle="" at="" 7.="">

Ans:
Let \(x\) = the fourth test score.
Write the average inequality: The average of four tests must be at least 85.
\[\frac{85 + 92 + 78 + x}{4} \geq 85\]
Simplify the numerator:
\[\frac{255 + x}{4} \geq 85\]
Multiply both sides by 4:
\[255 + x \geq 340\]
Subtract 255 from both sides:
\[x \geq 85\]
Check: If \(x = 85\): average = \(\frac{85 + 92 + 78 + 85}{4} = \frac{340}{4} = 85\) ✓
Final Answer: The student needs a score of at least 85 on the fourth test.

Ans:
Distribute the 3:
\[3x - 6 = 2x + 1\]
Subtract \(2x\) from both sides:
\[3x - 2x - 6 = 1\]
\[x - 6 = 1\]
Add 6 to both sides:
\[x = 7\]
Verify: Substitute \(x = 7\) back into the original equation.
Left side: \(3(7 - 2) = 3(5) = 15\)
Right side: \(2(7) + 1 = 14 + 1 = 15\)
Both sides equal 15, so the solution is correct. ✓
Final Answer: \(x = 7\)

Ans:
Claim: The student's claim is incorrect. When dividing both sides of an inequality by a negative number, the inequality sign must be reversed.
Evidence: The correct solution process is: divide both sides by \(-3\) and reverse the inequality sign to get \(x > -4\). We can verify this using a test value: if \(x = -3\) (which satisfies \(x > -4\)), then \(-3(-3) = 9 < 12\)="" is="" true,="" confirming="" that="" \(x="-3\)" is="" indeed="" a="" solution.="" however,="" if="" we="" use="" the="" student's="" incorrect="" answer="" of="" \(x="">< -4\),="" testing="" \(x="-5\)" gives="" \(-3(-5)="15">< 12\),="" which="" is="">false, showing this value should not be a solution.
Reasoning: The Inequality Reversal Property is essential because multiplying or dividing by a negative number reverses the order of numbers on a number line. For example, \(3 < 5\),="" but="" \(-3=""> -5\). Forgetting to reverse the inequality sign produces a solution set containing numbers that do not actually satisfy the original inequality, leading to mathematically incorrect conclusions.

Ans:
Claim: Plan A becomes the better deal when the number of calls exceeds 80 calls per month.
Evidence: Let \(c\) = number of calls per month. The cost equations are: Plan A: \(\text{Cost}_A = 30\) and Plan B: \(\text{Cost}_B = 10 + 0.25c\). To find when Plan A costs less, set up the inequality \(30 < 10="" +="" 0.25c\).="" solving:="" \(30="" -="" 10="">< 0.25c\),="" so="" \(20="">< 0.25c\),="" and="" dividing="" by="" 0.25="" gives="" \(c=""> 80\). At exactly 80 calls, both plans cost the same: \(\text{Cost}_B = 10 + 0.25(80) = 10 + 20 = 30\). Testing \(c = 81\): Plan A costs $30 and Plan B costs \(10 + 0.25(81) = 30.25\), confirming Plan A is cheaper.
Reasoning: This problem demonstrates how equations and inequalities model real-world situations. The solution shows that for customers who make more than 80 calls per month, Plan A's fixed $30 fee saves money compared to Plan B's accumulating per-call charges. Customers making fewer calls should choose Plan B. This comparison illustrates the practical application of solving inequalities to make informed financial decisions.

Ans:
Approach: This rational equation requires clearing denominators by finding a common denominator or using cross-multiplication.
Step 1 - Cross-multiply: Multiply both sides by the product of the denominators (5 and 2):
\[2 \cdot (2x + 3) = 5 \cdot (x - 1)\]
Step 2 - Distribute on both sides:
\[4x + 6 = 5x - 5\]
Step 3 - Collect variable terms on one side: Subtract \(4x\) from both sides:
\[6 = 5x - 4x - 5\]
\[6 = x - 5\]
Step 4 - Isolate \(x\): Add 5 to both sides:
\[11 = x\]
Step 5 - Verify the solution in the original equation:
Left side: \(\frac{2(11) + 3}{5} = \frac{22 + 3}{5} = \frac{25}{5} = 5\)
Right side: \(\frac{11 - 1}{2} = \frac{10}{2} = 5\)
Both sides equal 5. ✓
Step 6 - Check for restrictions: The original equation has denominators 5 and 2. Since neither denominator equals zero when \(x = 11\), there are no restrictions violated.
Final Answer: \(x = 11\)

Ans:
Approach: In a compound inequality with "and" (expressed as chaining), solve by applying the same operations to all three parts simultaneously.
Step 1 - Add 2 to all three parts:
\[-5 + 2 \leq 3x - 2 + 2 < 10="" +="" 2\]="">
\[-3 \leq 3x < 12\]="">
Step 2 - Divide all three parts by 3:
\[\frac{-3}{3} \leq \frac{3x}{3} < \frac{12}{3}\]="">
\[-1 \leq x < 4\]="">
Step 3 - Graph on a number line: Draw a closed circle at \(-1\) (because \(\leq\) includes \(-1\)) and an open circle at \(4\) (because \(<\) excludes="" \(4\)).="" shade="" the="" line="" segment="" between="" them.="">
Step 4 - Explain the difference from two separate inequalities: Solving as two separate inequalities would require:
Inequality 1: \(-5 \leq 3x - 2 \Rightarrow x \geq -1\)
Inequality 2: \(3x - 2 < 10="" \rightarrow="" x="">< 4\)="">
Then combining: \(x \geq -1\) AND \(x < 4\),="" which="" is="" \(-1="" \leq="" x="">< 4\).="">
Why chaining is more efficient: Solving as a chain allows us to perform each operation once on all three parts, rather than repeating operations on separate inequalities and then combining at the end. Both methods yield the same answer, but chaining reduces work and is less error-prone.
Step 5 - Verify with a test value: Let \(x = 0\) (in the solution set). Then \(3(0) - 2 = -2\). Check: \(-5 \leq -2 < 10\)?="" yes.="" ✓="">
Final Answer: \(-1 \leq x < 4\).="" graph:="" closed="" circle="" at="" \(-1\),="" open="" circle="" at="" \(4\),="" with="" shading="" between.="" chaining="" equations="" is="" more="" efficient="" because="" it="" reduces="" the="" number="" of="" operations="" and="" avoids="" the="" separate-then-combine="" step.="">