What Makes a Number Sentence True or False?

What Makes a Number Sentence True or False? exercise

• Identify the false number sentences.

  Hints

  First, simplify the given expressions.

  Next, decide if the relation becomes true.

  Just look at the following examples:

  • $13>12$
  • $13\ge12$
  • $13=13$

  There is only one true sentence.

  Solution

  Poor Ziggy. He lost his dog. Unfortunately, he has to solve different problems to turn the lights on so that he can see where his dog has gone:

  1. $4+6<9$? Is this true or false? First, we do addition on the left side of the relation sign: $4+6=10$. Sure, $10$ is not smaller than $9$ and thus this sentence is false.
  2. $3\times 6=18$? What about this sentence? Again we calculate the product $3\times 6$. The result is $18$. Thus this sentence is true.
  3. $22\ge 11+12$? Again we do addition first $11+12=23$. But $22$ isn't greater or equal to $23$. So this sentence is false too.

• Analyze the number sentences.

  Hints

  First, put $x=10$ into the sentence. For example, $10+x$ leads to $10+10=20$.

  Next, decide if the relation is true or false. For this, remember

  • $20>15$
  • $20\ge 20$
  • $20=20$

  Pay attention: $20\ge 20$, but $20\not > 20$.

  Solution

  Ziggy has to put $x=10$ in for each sentence to check the sentences.

  $\mathbf{25-x<15}$ leads to $25-10=15<15$? This is false because $15=15$ but not less than $15$.

  $\mathbf{13\ge x+4}$ leads to $13\ge 10+4=14$? But $13<14$ and so this sentence is false too.

  $\mathbf{\frac{70}x=7}$ leads to $\frac{70}{10}=7$? Sure, this is true.

• Determine which numbers make the number sentences true.

  Hints

  Put each value for $x$ in the equation first. Check the inequality after.

  Five values for $x$ which satisfy the inequality.

  There is only one value for $x$ which fulfills neither the equation nor the inequality.

  Solution

  Not again, Ziggy thinks. He has to solve another problem to get a coke and a taco for relaxation time.

  Let's start with the equation: $3\times x=15$. We check it with each given value for $x$.

  • $x=1$ $\rightarrow$ $3\times 1=3\not = 15$
  • $x=2$ $\rightarrow$ $3\times 2=6\not = 15$
  • $x=3$ $\rightarrow$ $3\times 3=9\not = 15$
  • $x=4$ $\rightarrow$ $3\times 4=12\not = 15$
  • $x=5$ $\rightarrow$ $3\times 5=15$ ✓
  • $x=6$ $\rightarrow$ $3\times 6=18\not = 15$

  So we only have to check $x=5$ because this $x$ must fulfill both the equation as well as the inequality: $5+4=9\le 9$ ✓

  Enjoy your relaxation time with your dogs, Ziggy.

• Find the mistakes.

  Hints

  First, calculate the given terms. For example $3\times 7=21$.

  Differentiate between $<$ and $\le$. Just look at the following example:

  • $12\le 12$ but
  • $12\not < 12$.

  $\le$ includes $=$. In the same way $\ge$ includes $=$.

  Solution

  The dogs are sleeping and Ziggy is still a little jumpy. What's the best you can do when you're jumpy: check relation sentences of course! So this is what Ziggy does.

  1. $3+7\le10$? First, he does the addition on the left side $3+7=10$. This sentence is true because $\le$ includes $=$.
  2. $3+7<10$ looks quite familiar. What's the difference? Here we have the $<$ sign and, sure, $10$ is not less than $10$. Thus this sentence is false.
  3. $x+7\le 10$. Here we have a variable $x$ for which we put $2$ in to get $2+7=9\le10$. This is a true sentence.
  4. $3\times 7>10$. Again we do the math first, i.e. $3\times 7=21$. $21>10$? Absolutely, this is a true sentence.
  5. $3\times 7-11>10$. This looks a little bit more complicated. First, we determine the left side to get: $3\times 7-11=21-11=10$. But $10$ isn't greater than $10$ itself. This sentence is false.
  6. $7-3\ge 5$? The difference is $7-3=4$ and $4$ is less than $5$. This sentence is false too.

• Categorize the different kinds of number sentences.

  Hints

  Look at the relation sign in the given sentence.

  You have to assign two sentences to each relation sign.

  $<$ means "less than" and $>$ means "greater than".

  Just read the sentences from left to the right and not the other way round.

  Solution

  In each of the following sentences the relation sign is highlighted, so you can recognize it directly:

  1. $4+6 \mathbf{\color{#669900}{<}} 9$
  2. $25-x \mathbf{\color{#669900}{<}} 15$
  3. $22 \mathbf{\color{#669900}{>}} 11+12$
  4. $13 \mathbf{\color{#669900}{>}} x+4$
  5. $3\times 6 \mathbf{\color{#669900}{=}} 18$
  6. $70\div x\mathbf{\color{#669900}{=}} 7$

• Determine the relation sign to make the number sentence true.

  Hints

  The relation sign isn't unique. You can use different ones to make the sentence true.

  Just look at the following example:

  $3$ ___ $4$.

  Here you can put in $<$ as well as $\le$ in place of the ___ to make this sentence true.

  Simplify the terms as much as possible first.

  Solution

  First, we have $3\times (5+4)$ ___ $30-6$:

  Simplify the given expression using PEMDAS:

  • $3\times (5+4)=3\times )=27$
  • $30-6=24$

  Thus we have $27>24$ or $27\ge 24$.

  Next, we have $3\times 5+4$ ___ $60\div 2$:

  • $3\times 5+4=15+4=19$
  • $60\div 2=20$

  So we get $19<20$ or $19\le 20$.

  Then we have $3\times 5-6$ ___ $27\div(2+1)$

  • $3\times 5-6=15-6=9$
  • $27\div(2+1)=27\div 3=9$

  Together we can conclude $9=9$ as well as $9\le 9$ as well as $9\ge 9$.

  Finally, we have $3\times 5-6$ ___ $27\div(3\times 3)$:

  • $3\times 5-6=15-6=9$
  • $27\div(3\times 3)=27\div9=3$

  This gives us $9>3$ as well as $9\ge 3$.