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<p>By solving the pair of inequalities $3(a-5)<2a-3$ and $2-a\le 5a$, find</p>
<ol class="i-list"><li><div class="list-item-paragraph">the least possible value of $a$,</div></li><li><div class="list-item-paragraph">the greatest possible integer value of $a$,</div></li><li><div class="list-item-paragraph">the greatest possible perfect square value of $a$.</div></li></ol>

By solving the pair of inequalities $3(a-5)<2a-3$ and $2-a\le 5a$, find

Solving Simultaneous Linear Inequalities

Solving Simultaneous Inequalities and Identifying Special Values

Inequalities

<ol class="i-list"><li><div class="list-item-paragraph">Solve the inequality $x-1<\frac{2x+1}{3}\le x+\frac{4}{3}$.</div></li><li><div class="list-item-paragraph">Write down the number of integers that satisfy the inequality in <strong>(a).</strong></div></li></ol>

Finding the Number of Integer Values Satisfying a Compound Inequality

<ol class="i-list"><li><div class="list-item-paragraph">Solve the inequality $4x-7<\frac{3x-1}{4}\le 5$.</div></li><li><div class="list-item-paragraph">State the largest prime number which satisfies the above inequality $4x-7<\frac{3x-1}{4}\le 5$.</div></li></ol>

Double Inequality with Rational Expressions and Prime Constraints

<p>Solve the inequality $\frac{x+2}{4}\le \frac{2x-1}{3}<3\frac{1}{2}$ and hence, find the largest prime number value of $x$ which satisfies the inequality.</p>

Solve the inequality $\frac{x+2}{4}\le \frac{2x-1}{3}<3\frac{1}{2}$ and hence, find the largest prime number value of $x$ which satisfies the inequality.

Three-Part Inequality with Rational Expressions and Prime Constraints

<p>Solve the inequality $\frac{a}{2}-3<\frac{2a-5}{3}\le 10-a$ and show your solution on a number line.</p>

Solve the inequality $\frac{a}{2}-3<\frac{2a-5}{3}\le 10-a$ and show your solution on a number line.

Three-Part Linear Inequality with Number Line Sketch

<p>Solve the inequality $\frac{2-x}{3}\le 3+x<4$ and represent your solution on a number line.</p>

Solve the inequality $\frac{2-x}{3}\le 3+x<4$ and represent your solution on a number line.

Linear Fractional Inequality with Graphical Representation

<p>Find the range of values of $x$ that satisfies $3x+1>2$ and $2x-10<5-x$.</p>

Find the range of values of $x$ that satisfies $3x+1>2$ and $2x-10<5-x$.

Finding the Range of Values Satisfying Multiple Inequalities

<ol class="i-list"><li><div class="list-item-paragraph">Solve the inequality $-2\le 2x+4<17$.</div></li><li><div class="list-item-paragraph">Show your solution on a number line.</div></li><li><div class="list-item-paragraph">Find the smallest prime number that satisfy the inequality $-2\le 2x+4<17$.</div></li></ol>

Solving a Compound Linear Inequality and Identifying Prime Solutions

<p>Given that $x$ and $y$ are integers and $2\le x\le 5$ and $-8\le y\le -3$, find</p>
<ol class="i-list"><li><div class="list-item-paragraph">the least possible value of ${{x}^{2}}+y$,</div></li><li><div class="list-item-paragraph">the largest possible value of $x-y$,</div></li><li><div class="list-item-paragraph">the largest possible value of $\frac{x}{y}$.</div></li></ol>

Given that $x$ and $y$ are integers and $2\le x\le 5$ and $-8\le y\le -3$, find

Finding Extremes of Algebraic Expressions Within Integer Constraints

<p>It is given that $\frac{3x-1}{4}\le \frac{7x+4}{3} < x+2\frac{2}{3}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Solve the inequality.</div></li><li><div class="list-item-paragraph">Hence, list the integer values of $x$ that satisfy the inequality.</div></li></ol>

It is given that $\frac{3x-1}{4}\le \frac{7x+4}{3} < x+2\frac{2}{3}$.

Solving a Three-Part Compound Inequality Involving Algebraic Fractions

<ol class="i-list"><li><div class="list-item-paragraph">Solve the inequalities $-10\le 7-2x<1$.</div></li><li><div class="list-item-paragraph">Write down all the prime numbers that satisfy $-10\le 7-2x<1$.</div></li></ol>

Solving and Interpreting a Compound Inequality with Prime Number Constraints

<ol class="abc-list"><li><div class="list-item-paragraph">Find the range of values of $x$ for which</div><ol class="i-list"><li><div class="list-item-paragraph">$x-1>1-x$.</div></li><li><div class="list-item-paragraph">$6\le 10-x\le 10$</div></li></ol></li><li><div class="list-item-paragraph">Represent the solution to both inequalities in <strong>(a)(i)</strong> and <strong>(a)(ii)</strong> on a number line.</div></li><li><div class="list-item-paragraph">Hence, state the least integer that satisfies both inequalities in <strong>(a)(i)</strong> and <strong>(a)(ii)</strong>.</div></li></ol>

Solving and Interpreting a System of Inequalities with Number Line Representation

<p>In some cases, an unknown $x$ has to satisfy two or more linear inequalities simultaneously. </p>
<p>Then the solution will be the <strong>common solution</strong> to all the simultaneous linear inequalities. </p>
<p>Let’s try to find the common solution for the following pairs of inequalities by first representing each inequality on the same number line and then find the overlapping region. </p>
<ol class="abc-list"><li><div class="list-item-paragraph">$x>2$ and $x\le 4$</div><div class="list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1765530832114-hch72ugasg.png" alt="List item image" class="aligncenter custom-width" style="max-width: 400px; height: auto;" /></div></li><li><div class="list-item-paragraph">$x<-1$ and $x\le 2$</div><div class="list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1765530882817-thw7lf04qss.png" alt="List item image" class="aligncenter custom-width" style="max-width: 400px; height: auto;" /></div><div class="list-item-paragraph" style="text-align: center">Solution: $x<-1$</div></li></ol>
<p>The inequalities that represent the overlapping region is the common solution.</p>

In some cases, an unknown $x$ has to satisfy two or more linear inequalities simultaneously. 

Then the solution will be the **common solution** to all the simultaneous linear inequalities. 

Let’s try to find the common solution for the following pairs of inequalities by first representing each inequality on the same number line and then find the overlapping region. 

The inequalities that represent the overlapping region is the common solution.

Common Solution

<p>To receive a grade B for a course, a candidate must have a mean score of at least $80$ but less than $90$ in three examination papers.  </p>
<p>The full score in each paper is $100$.</p>
<p>A student scored $83$ and $75$ in the first two papers.  </p>
<p>What range of score should this student have in the third paper in order to get a grade B for the course?</p>

To receive a grade B for a course, a candidate must have a mean score of at least $80$ but less than $90$ in three examination papers.  

A student scored $83$ and $75$ in the first two papers.  

What range of score should this student have in the third paper in order to get a grade B for the course?

Problem Solving Involving Inequalities

Forming and Solving a Compound Inequality from a Grade Requirement

<p>A newspaper company charges $25$ cents per word and a fixed charge of $80$ cents for and advertisement.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Calculate the total cost of an advertisement containing $15$ words.</div><div class="list-item-paragraph">Leave your answer in dollars.</div></li><li><div class="list-item-paragraph">The total cost of an advertisement containing $n$ words is $C$ cents.  </div><div class="list-item-paragraph">Find a formula for&nbsp;$C$ in terms of $n$.</div></li><li><div class="list-item-paragraph">Arthur does not want to spend more than $\$5.00$ on an advertisement.</div><div class="list-item-paragraph">What is the greatest number of words he can use?</div></li></ol>

A newspaper company charges $25$ cents per word and a fixed charge of $80$ cents for and advertisement.

Forming and Solving an Inequality from a Word Count Pricing Scenario

<p>Company $X$ quotes a car rental rate of $\$65$ per day.</p>
<p>Company $Y$ quotes a rate of $\$56$ per day with an additional initial charge of $\$86.50$.  </p>
<p>Let $x$ be the number of days to rent a car.</p>
<p>Form an inequality in $x$ and use it to find the minimum number of complete days required such that charges of Company $Y$ will become cheaper than Company $X$.</p>

Company $X$ quotes a car rental rate of $\$65$ per day.

Company $Y$ quotes a rate of $\$56$ per day with an additional initial charge of $\$86.50$.  

Let $x$ be the number of days to rent a car.

Form an inequality in $x$ and use it to find the minimum number of complete days required such that charges of Company $Y$ will become cheaper than Company $X$.

Comparing Rental Costs Using Inequalities: Finding the Minimum Number of Days

<p>The table below shows the cost of desks and chairs from a Furniture Shop.</p>
<div class="table-container"><table><colgroup><col style="width: 200px;"><col style="width: 100px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p>Items</p></th><th style="text-align: center; --cell-justify: center"><p>Cost</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>Student Desk</p></td><td style="text-align: center; --cell-justify: center"><p>$\$30$ per item</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>Student Chair</p></td><td style="text-align: center; --cell-justify: center"><p>$\$25$ per item</p></td></tr></tbody></table></div>
<p>A tuition centre would like to purchase $20$ desks and as many chairs as possible given a budget of $\$1,200$.  </p>
<p>There is a delivery charge of $\$30$.</p>
<p>Using an inequality, calculate the maximum number of chairs she can purchase.</p>

The table below shows the cost of desks and chairs from a Furniture Shop.

A tuition centre would like to purchase $20$ desks and as many chairs as possible given a budget of $\$1,200$.  

Using an inequality, calculate the maximum number of chairs she can purchase.

Furniture Purchase Problem: Solving a Budget Inequality

<p>A Telco company charges $\$5$ for the first $10$ minutes of international call and $90$ cents per minute subsequently.</p>
<p>On a particular month, Kenny travelled overseas for work.  </p>
<p>The bill for his international call was not more than $\$20.25$ for the month.</p>
<p>Form an inequality to find the maximum number of minutes Kenny spent on international call for the month.</p>

A Telco company charges $\$5$ for the first $10$ minutes of international call and $90$ cents per minute subsequently.

On a particular month, Kenny travelled overseas for work.  

The bill for his international call was not more than $\$20.25$ for the month.

Form an inequality to find the maximum number of minutes Kenny spent on international call for the month.

Forming an Inequality from a Real-World Cost Scenario Involving Time and Money

<p>John scored $38$, $32$, $49$ and $x$ marks for four Mathematics tests.  </p>
<p>He must score an average of at least $50$ marks if he wants to obtain a pass in Mathematics.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Form an inequality in $x$ for the information above so that John can pass his Mathematics.</div></li><li><div class="list-item-paragraph">Solve the inequality.</div></li><li><div class="list-item-paragraph">Hence, state the minimum mark that John must score for the last Mathematics test in order to pass.</div></li></ol>

John scored $38$, $32$, $49$ and $x$ marks for four Mathematics tests.  

He must score an average of at least $50$ marks if he wants to obtain a pass in Mathematics.

Forming and Solving an Inequality from an Average Score Requirement

<p>Alicia and Betty planned for a party and they decided not to spend more than $\$150$. Alicia spent $\$25$ more than one-fourth of what Betty spent.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Let the amount that Betty spent be $\$x$, form an inequality and do not simplify.</div></li><li><div class="list-item-paragraph">Solve the inequality to find the maximum amount that Betty spent.</div></li></ol>

Alicia and Betty planned for a party and they decided not to spend more than $\$150$. Alicia spent $\$25$ more than one-fourth of what Betty spent.

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