Solving Inequalities

Solving Inequalities exercise

• What does each inequality mean?

  Hints

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

  $\leq$ means less than or equal to and $\geq$ means greater than or equal to.

  We can write x is greater than 9 as $x > 9$ or $9 < x$.

  Solution

  $x > 2$ means $x$ is greater than $2$

  $x \leq 4$ means $x$ is less than or equal to $4$

  $x \geq 4$ means $x$ is greater than or equal to $4$

  $x < 2$ means $x$ is less than $2$

  $5 > x$ means $x$ is less than $5$

  $5 < x$ means $x$ is greater than $5$

• True or False?

  Hints

  To test the inequality, replace the variable (the letter) by the number.

  For example, does $a = 5$ satisfy the inequality $a < 7$?

  Replace $a$ with $5$, giving $5 < 7$.

  $5$ is less than $7$ so this is true.

  You can use a number line to help you. Numbers to the left are less than numbers to the right.

  In the last example, $5 < 7$ because $5$ is to the left of $7$.

  We can also say that, $7 > 5$ because $7$ is to the right of $5$.

  Be careful with negative numbers. For example $-4$ is less than $-1$ as $-4$ is to the left of $-1$.

  Solution

  The inequality $x > 5$, is true when $x$ is $9$?

  The inequality $y < 2$, is false when $y = 4$?

  When $m > 8$, it is false that $m = 8$ satisfies the inequality?

  The inequality $t \geq -5$ is true when $t$ is $-3$?

  When $p \leq 7$, it is true that $p$ is $7$ satisfies the equation?

• Solve an inequality with a negative variable.

  Hints

  Solving inequalities is often similar to solving equations.

  Remember to reverse the inequality sign when dividing by a negative.

  For example: $-5t > 35$

  Divide by $-5$ and reverse the inequality giving:

  $t < -7$

  Solution

  $-6p - 3 > 15$

  Add $3$ to both sides.

  $-6p > 18$

  Divide both sides by $-6$, remember to reverse the inequality sign when dividing by a negative.

  $p < -3$

  Result: p is less than $-3$.

• Solve the inequality.

  Hints

  Solving inequalities is often similar to solving equations.

  If the inequality has variables on both sides, we start by addressing this. Remove the smaller variable first by carrying out the inverse operation.

  For example: $7y - 6 \geq 2y + 4$.

  Step one would be to subtract $2y$ on both sides.

  This gives $5y - 6 \geq 4$.

  Solution

  Solve $5m - 8 \leq 3m + 4$

  Step one: Subtract $3m$ on both sides.

  $\bf{2m}$ $- 8 \leq 4$

  Step two: Add $\bf{8}$ to both sides.

  $\bf{2m \leq 12}$

  Step three: Divide both sides by $\bf{2}$.

  $m \leq$ $\bf{6}$

  Result: $m$ is less than or equal to $\bf{6}$.

• Solve the inequality.

  Hints

  Start with the inequality you are solving.

  Remember to state the result at the end. This means that you write the inequality in words.

  For example, $t < 7$ means $t$ is less than $7$.

  Solution

  Start with the inequality:

  $2x + 9 < 15$

  Subtract 9 from each side:

  $2x$ $< 6$

  Divide both sides by 2:

  $x$ $< 3$

  State the result:

  $x$ is less than 3.

• Solve an inequality with a bracket.

  Hints

  It is helpful to expand the bracket first.

  $x$ and $1x$ are equivalent.

  Remember to use inverse operations to solve the inequality.

  Solution

  Solve $5x + 10 \geq 4(x + 3)$

  $5x + 10 \geq 4x + 12$

  $1x + 10 \geq 12$

  $x \geq 2$